Explanation:
Okay, so I need to figure out how many ways 5 students can be selected from a group of 6 students. Hmm, let's start by recalling some basic combinatorics principles.

First off, when the problem is about selecting a subset from a larger set without considering the order, that's a combination problem, right? Because if the order mattered, it would be a permutation. But here, it's just about selecting 5 out of 6, so order doesn't matter.

So combinations are calculated using the formula C(n, k) = n! / (k! * (n - k)!), where n is the total number of items, and k is the number of items to choose. Let me make sure I remember that correctly. Yeah, combinations are about grouping without regard to order.

In this case, n is 6 students, and k is 5 students. So plugging into the formula, that would be C(6, 5) = 6! / (5! * (6 - 5)!). Let me compute that step by step.

First, compute 6! which is 6 factorial. 6! = 6 × 5 × 4 × 3 × 2 × 1. But wait, maybe there's a shortcut here because we have 5! in the denominator. Let me think.

The formula can be simplified by expanding the factorials. Let's do that.

C(6, 5) = 6! / (5! * 1!)

Because 6 - 5 is 1, so the denominator is 5! times 1!.

Now, 6! is 6 × 5!, right? Because 6! = 6 × 5 × 4 × 3 × 2 × 1 = 6 × 5!. So substituting that into the equation:

C(6, 5) = (6 × 5!) / (5! × 1!)

Now, I can cancel out the 5! in the numerator and denominator. That leaves:

C(6, 5) = 6 / 1!

Since 1! is 1, this simplifies to 6 / 1 = 6.

Wait, so the number of ways is 6? Let me check if that makes sense.

Another way to think about it: if there are 6 students and I need to choose 5, that's the same as leaving out 1 student. So the number of ways to choose 5 students is equal to the number of ways to leave out 1 student.

Since there are 6 students, there are 6 different students who could be the one left out. Therefore, there are 6 ways to choose 5 students.

Yes, that matches the previous calculation. So both methods give me 6.

Alternatively, maybe I can think about permutations. If I were to arrange 5 students out of 6, the number of permutations would be P(6, 5) = 6 × 5 × 4 × 3 × 2. But since order doesn't matter in combinations, each combination is counted multiple times in permutations. Specifically, each group of 5 students can be arranged in 5! ways.

So the number of combinations would be P(6, 5) divided by 5!.

Calculating that: P(6, 5) = 6 × 5 × 4 × 3 × 2 = 720.

Then divide by 5! which is 120.

720 / 120 = 6.

Same answer again.

So regardless of the method I use—direct combination formula, thinking about leaving out one student, or using permutations and adjusting for overcounting—the result is 6.

I think that's solid. Let me just check with a smaller example to make sure my reasoning isn't flawed.

Suppose there are 3 students and I want to choose 2. How many ways?

Using combinations: C(3, 2) = 3! / (2! * 1!) = 6 / (2 * 1) = 3.

Alternatively, leaving out one student: there are 3 students, so leaving out 1 student gives 3 ways.

Yes, that works. The combinations are {A,B}, {A,C}, {B,C}, which is 3.

Another check: if I have 4 students and choose 3.

C(4, 3) = 4! / (3! * 1!) = 24 / (6 * 1) = 4.

Alternatively, leaving out one student: 4 ways. That also checks out.

So, applying the same logic, with 6 students choosing 5, it's 6.

I think that's correct.

But wait, just to make sure I didn't confuse combinations with permutations. If the question had said arranging 5 students out of 6, then it would be permutations. But since it's just selecting, combinations are the way to go.

Another way to think about combinations: the number of combinations C(n, k) is equal to C(n, n - k). So C(6, 5) is equal to C(6, 1). Which is 6.

Yes, that's a helpful identity. So choosing 5 out of 6 is the same as choosing 1 out of 6 to exclude. Hence, 6 ways.

Therefore, all methods converge to the answer 6.

I think that's thorough. I can't think of any other way this could be approached, and all the different methods give me the same answer.

**Final Answer**
The number of ways is \boxed{6}.

To determine how many ways 5 students can be selected from a group of 6, we approach the problem as a **combination** question. This is because the order in which we select the students does **not** matter — only the group that ends up being selected is important.

---

### Understanding the Situation

We're choosing 5 students out of 6 available. This can be represented using the **binomial coefficient** (also called combinations):

$$
\binom{n}{k} = \frac{n!}{k!(n-k)!}
$$

Where:
- $ n $ = total number of items (students) = 6
- $ k $ = number of items to choose (students to select) = 5

So, the number of ways to choose 5 students out of 6 is:

$$
\binom{6}{5} = \frac{6!}{5!(6-5)!} = \frac{6!}{5! \cdot 1!}
$$

We can simplify this expression by noting that:

- $ 6! = 6 \times 5! $

So:

$$
\frac{6 \times 5!}{5! \cdot 1!} = \frac{6}{1} = 6
$$

---

### Alternative Perspective

Another way to think about this problem is by focusing on what is **left out** rather than what is selected. Since we're selecting 5 students from 6, that means only **one** student is left out.

- There are 6 students in total.
- Each student could potentially be the one left out.
- Therefore, there are **6** different groups of 5 students that can be formed by omitting one student each time.

This matches the earlier calculation and provides further confirmation that the answer is **6**.

---

### Final Answer

$$
\boxed{6}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many ways 5 students can be selected from a group of 6 students. Hmm, let's start by recalling some basic combinatorics principles.

First off, when the problem is about selecting a subset from a larger set without considering the order, that's a combination problem, right? Because if the order mattered, it would be a permutation. But here, it's just about selecting 5 out of 6, so order doesn't matter.

So combinations are calculated using the formula C(n, k) = n! / (k! * (n - k)!), where n is the total number of items, and k is the number of items to choose. Let me make sure I remember that correctly. Yeah, combinations are about grouping without regard to order.

In this case, n is 6 students, and k is 5 students. So plugging into the formula, that would be C(6, 5) = 6! / (5! * (6 - 5)!). Let me compute that step by step.

First, compute 6! which is 6 factorial. 6! = 6 × 5 × 4 × 3 × 2 × 1. But wait, maybe there's a shortcut here because we have 5! in the denominator. Let me think.

The formula can be simplified by expanding the factorials. Let's do that.

C(6, 5) = 6! / (5! * 1!)

Because 6 - 5 is 1, so the denominator is 5! times 1!.

Now, 6! is 6 × 5!, right? Because 6! = 6 × 5 × 4 × 3 × 2 × 1 = 6 × 5!. So substituting that into the equation:

C(6, 5) = (6 × 5!) / (5! × 1!)

Now, I can cancel out the 5! in the numerator and denominator. That leaves:

C(6, 5) = 6 / 1!

Since 1! is 1, this simplifies to 6 / 1 = 6.

Wait, so the number of ways is 6? Let me check if that makes sense.

Another way to think about it: if there are 6 students and I need to choose 5, that's the same as leaving out 1 student. So the number of ways to choose 5 students is equal to the number of ways to leave out 1 student.

Since there are 6 students, there are 6 different students who could be the one left out. Therefore, there are 6 ways to choose 5 students.

Yes, that matches the previous calculation. So both methods give me 6.

Alternatively, maybe I can think about permutations. If I were to arrange 5 students out of 6, the number of permutations would be P(6, 5) = 6 × 5 × 4 × 3 × 2. But since order doesn't matter in combinations, each combination is counted multiple times in permutations. Specifically, each group of 5 students can be arranged in 5! ways.

So the number of combinations would be P(6, 5) divided by 5!.

Calculating that: P(6, 5) = 6 × 5 × 4 × 3 × 2 = 720.

Then divide by 5! which is 120.

720 / 120 = 6.

Same answer again.

So regardless of the method I use—direct combination formula, thinking about leaving out one student, or using permutations and adjusting for overcounting—the result is 6.

I think that's solid. Let me just check with a smaller example to make sure my reasoning isn't flawed.

Suppose there are 3 students and I want to choose 2. How many ways?

Using combinations: C(3, 2) = 3! / (2! * 1!) = 6 / (2 * 1) = 3.

Alternatively, leaving out one student: there are 3 students, so leaving out 1 student gives 3 ways.

Yes, that works. The combinations are {A,B}, {A,C}, {B,C}, which is 3.

Another check: if I have 4 students and choose 3.

C(4, 3) = 4! / (3! * 1!) = 24 / (6 * 1) = 4.

Alternatively, leaving out one student: 4 ways. That also checks out.

So, applying the same logic, with 6 students choosing 5, it's 6.

I think that's correct.

But wait, just to make sure I didn't confuse combinations with permutations. If the question had said arranging 5 students out of 6, then it would be permutations. But since it's just selecting, combinations are the way to go.

Another way to think about combinations: the number of combinations C(n, k) is equal to C(n, n - k). So C(6, 5) is equal to C(6, 1). Which is 6.

Yes, that's a helpful identity. So choosing 5 out of 6 is the same as choosing 1 out of 6 to exclude. Hence, 6 ways.

Therefore, all methods converge to the answer 6.

I think that's thorough. I can't think of any other way this could be approached, and all the different methods give me the same answer.

**Final Answer**
The number of ways is \boxed{6}.

To determine how many ways 5 students can be selected from a group of 6, we approach the problem as a **combination** question. This is because the order in which we select the students does **not** matter — only the group that ends up being selected is important.

---

### Understanding the Situation

We're choosing 5 students out of 6 available. This can be represented using the **binomial coefficient** (also called combinations):

$$
\binom{n}{k} = \frac{n!}{k!(n-k)!}
$$

Where:
- $ n $ = total number of items (students) = 6
- $ k $ = number of items to choose (students to select) = 5

So, the number of ways to choose 5 students out of 6 is:

$$
\binom{6}{5} = \frac{6!}{5!(6-5)!} = \frac{6!}{5! \cdot 1!}
$$

We can simplify this expression by noting that:

- $ 6! = 6 \times 5! $

So:

$$
\frac{6 \times 5!}{5! \cdot 1!} = \frac{6}{1} = 6
$$

---

### Alternative Perspective

Another way to think about this problem is by focusing on what is **left out** rather than what is selected. Since we're selecting 5 students from 6, that means only **one** student is left out.

- There are 6 students in total.
- Each student could potentially be the one left out.
- Therefore, there are **6** different groups of 5 students that can be formed by omitting one student each time.

This matches the earlier calculation and provides further confirmation that the answer is **6**.

---

### Final Answer

$$
\boxed{6}
$$

Solution: We can choose 5 students out of a group of 6 students  without regard to order in $\binom{6}{5} = \boxed{6}$ ways.


Explanation:
Okay, so Linda, Sherry, June, and Connie sold cookies and earned different amounts of money. Now they want to pool their money together, convert coins into dollars, and figure out how much money in cents is left after they convert as many coins as possible into bills. Hmm, let me think about how to approach this.

First, I need to figure out the total amount of money they have. Let me write down each person's earnings:

- Linda: $27.47
- Sherry: $35.23
- June: $37.37
- Connie: $26.52

So, adding these up. Let me do that step by step. Let's start with Linda and Sherry. 27.47 plus 35.23. Let me add the dollars and cents separately. 27 + 35 is 62, and 47 cents + 23 cents is 70 cents. So that's $62.70 together.

Then June and Connie: 37.37 + 26.52. Again, dollars and cents. 37 + 26 is 63, and 37 + 52 is 89 cents. So that's $63.89.

Now adding those two totals together: $62.70 + $63.89. Let's do that. 62 + 63 is 125, and 70 cents + 89 cents is 159 cents. 159 cents is $1.59. So adding that to $125 gives $126.59 total. Wait, let me check that again. Because 62.70 + 63.89. Let me add the decimal parts first: 0.70 + 0.89 = 1.59. Then the whole dollars: 62 + 63 = 125. So total is 125 + 1.59 = $126.59. Yes, that seems right.

So total money they have is $126.59. Now, the problem says they pooled their money together and went to the bank to convert the coins into dollars. How much money in cents was left after they converted as many of the coins into bills as they could?

Wait, so they have a certain amount of money, which is $126.59. Now, they want to convert as many coins into bills as possible. So, this is about converting coins to bills, so the leftover would be the coins that couldn't be converted into whole dollars?

But wait, the question is a bit confusing. Wait, when you convert coins into bills, you can exchange coins for bills. But if you have, for example, a certain amount of coins, you can exchange them for bills. However, the problem says "how much money, in cents, was left after they converted as many of the coins into bills as they could."

Wait, hold on. Maybe the initial amounts they earned include both coins and bills? Wait, but the problem says they earned amounts in dollars, like $27.47, which is dollars and cents. But when they pooled their money together, maybe they have a total amount of money in dollars and cents, and then they wanted to convert all coins into bills. However, the problem says "converted the coins into dollars". So, perhaps they have coins, and they want to exchange coins for dollar bills. But how does that work?

Wait, maybe they have all their money as coins, and they want to convert them into dollars. But wait, the problem says they "pooled their money together and went to the bank to convert the coins into dollars." So perhaps the question is about converting the coins into bills, meaning exchanging coins for paper money, and they want to know how much is left in coins after converting as much as possible to bills. But if they have total money, which is $126.59, which is in dollars and cents, then to convert coins into dollars, they would take the coins and exchange them for dollar bills. But if they have, say, a certain amount of coins, maybe the total amount they have is in coins. But the original problem says they earned amounts in dollars, which could be in bills and coins. Wait, this is confusing.

Wait, maybe the problem is that they have all their money in coins, and they want to convert as many coins as possible into bills, so the leftover is the coins that cannot be converted into whole dollars. But the total amount is $126.59. But if they have that amount in coins, then converting into bills would be taking the coins and exchanging them for bills. But how?

Alternatively, maybe the problem is that they have a certain amount of cents, and they want to convert as many as possible into dollars, so the leftover would be the cents left after dividing by 100. Wait, but the total amount is in dollars. Wait, the original total is $126.59, which is 126 dollars and 59 cents. If they convert all the coins into bills, they would take the cents and convert them into dollars? No, that doesn't make sense. Wait, maybe they have all their money in coins, which they want to convert into dollars. But the problem says they "pooled their money together and went to the bank to convert the coins into dollars." So, maybe they have all their money in coins, and they want to convert them into dollar bills. But how would that work? For example, if you have coins, you can exchange them for bills. But the question is about the leftover coins after converting as much as possible. However, if they convert all the coins into bills, there would be no coins left. That seems conflicting.

Wait, maybe the problem is actually about the cents left after converting the total amount into dollars and cents. Wait, but the question says "convert the coins into dollars," so perhaps the total amount they have is in cents. Wait, but the original amounts were in dollars. Wait, maybe the problem is that when they pooled their money together, they had a total amount in cents, and they wanted to convert as much as possible into dollars (i.e., take cents and convert them into dollars), leaving some cents. But that doesn't make sense because converting cents into dollars would mean giving up coins for bills. Wait, but if they have the money in cents, then converting into dollars would be taking the total cents and converting into dollars, so the leftover would be the remaining cents. Wait, maybe the problem is that they have a total amount of money, which is in dollars and cents, and the question is asking about the cents left after converting the coins into dollars. Wait, but how do we know how much was in coins vs. bills?

Hold on, maybe the key here is that when you have money, it's made up of bills and coins. But the problem says they pooled their money together and went to the bank to convert the coins into dollars. So maybe they have some number of coins, and they want to exchange them for dollar bills. But if they have, say, $126.59 in total, which is 126 dollars and 59 cents. If they want to convert the coins into dollars, that implies that they have the 59 cents in coins, and they want to convert those into dollars. But you can't convert cents into dollars unless you have more than a dollar. Wait, this is confusing.

Alternatively, maybe the problem is that when they had the money, they had it in coins, and they wanted to convert them into bills. So, for example, if they have coins, they can exchange them for bills. But when you exchange coins into bills, you can exchange any number of coins into bills. But if they want to convert as many coins as possible into bills, then they would take all the coins and exchange them for the maximum number of bills, and the leftover would be the coins that couldn't be converted. But if they have, for example, 59 cents, which is less than a dollar, then they can't convert that into a dollar bill. Therefore, the leftover would be 59 cents. Wait, but that seems too straightforward. However, the total amount is $126.59. If they have all that as coins, then converting as much as possible into bills would be converting all the whole dollars into bills and leaving the 59 cents as coins. Therefore, the leftover would be 59 cents. But that answer seems too easy, and maybe I'm missing something here.

Wait, but the problem says they "converted the coins into dollars". So maybe they have some number of coins, and they try to convert them into dollar bills. But the problem is that you can't convert less than a dollar into a bill. Therefore, the leftover would be the cents that couldn't be converted into a dollar. However, if they have, for example, $126.59 total, that is 126 dollars and 59 cents. If they have the 126 dollars as bills and the 59 cents in coins, then converting the coins into dollars... you can't convert 59 cents into a dollar bill. Therefore, they can't convert any coins into dollars, so the leftover is 59 cents. But that would be the answer. But why would the problem mention all their earnings? Maybe the problem is not about converting the cents, but something else?

Wait, perhaps the problem is that when they pooled their money together, they had a total amount in cents, and they wanted to convert as many cents as possible into dollars, leaving some cents. But if that's the case, then we need to convert the total amount into cents, then divide by 100 to find how many dollars they can convert, and the remainder would be the leftover cents. But the problem says "convert the coins into dollars", which might imply that the total amount is in coins. But how do we know how much was in coins?

Wait, perhaps the problem is that when they earned money, they received it in coins, so each of their amounts is in cents, and they want to convert as many as possible into dollars. Wait, but the problem says they earned amounts in dollars, like $27.47, which is dollars and cents. So if we convert each amount into cents, we can get the total number of cents they had. Then, convert as many as possible into dollars (which would be dividing the total cents by 100) and the remainder is the leftover cents. So maybe that's the case.

Let me try this approach. So, first convert each person's earnings into cents:

Linda: $27.47 = 27*100 + 47 = 2700 + 47 = 2747 cents

Sherry: $35.23 = 35*100 + 23 = 3523 cents

June: $37.37 = 37*100 + 37 = 3737 cents

Connie: $26.52 = 26*100 + 52 = 2652 cents

Total cents = 2747 + 3523 + 3737 + 2652.

Let me compute that step by step:

First, 2747 + 3523. Let's add 2747 + 3523. 2700 + 3500 = 6200. 47 + 23 = 70. So total is 6200 + 70 = 6270 cents.

Then, 3737 + 2652. Let's compute that. 3700 + 2600 = 6300. 37 + 52 = 89. So total is 6300 + 89 = 6389 cents.

Now, total cents is 6270 + 6389. Let's compute that.

6270 + 6389. Let me add 6270 + 6000 = 12270, then add 389: 12270 + 389 = 12659 cents.

So total cents is 12,659 cents. So if they converted as many as possible into dollars, how many dollars is that? Well, dividing 12659 by 100 gives 126 dollars with a remainder of 59 cents. Therefore, the leftover is 59 cents. Therefore, the answer is 59 cents. So, $\boxed{59}$.

Wait, but this is the same answer as before. So whether you think of the total amount as $126.59, which is 126 dollars and 59 cents, and thus leftover is 59 cents. But why did they mention all the individual amounts? Because if you just added up the total, you get $126.59, which is equivalent to 12659 cents, so converting into dollars gives 126 dollars and 59 cents, so leftover is 59 cents. But maybe there is a trick here?

Wait, but is there a possibility that they have coins and bills, and converting coins into dollars would involve exchanging coins for bills, but if they have some coins, they can exchange them. Wait, but if they have $126.59 in total, and they have some combination of bills and coins. For example, they may have a certain number of coins, and they want to exchange those coins into bills. However, unless we know how many coins they have, we can't determine the leftover. But the problem says "converted the coins into dollars." So, perhaps the question is assuming that all the money is in coins. Therefore, converting as much as possible into dollars would mean taking all the coins and exchanging them for dollars, which would leave you with the leftover cents. Therefore, since total cents is 12659, the leftover is 59 cents. Therefore, answer is 59.

Alternatively, maybe the question is about converting the coins into dollar bills, meaning exchanging coins for dollar bills, but if you have, for example, 126 dollars and 59 cents, the 126 dollars is already in bills, so you can't convert that into more bills. Whereas the 59 cents is in coins, so you can't convert that into a bill. Therefore, leftover is 59 cents. So regardless of interpretation, the answer seems to be 59 cents.

But why would the problem give all the individual amounts? Because if you just take the total amount, $126.59, and convert into cents, you get 12659 cents. Dividing by 100, the remainder is 59. Therefore, 59 cents left. But maybe there's another interpretation. Maybe they had coins, and when they went to the bank, they wanted to exchange coins for bills, but they had different denominations. Wait, but unless we know the exact number of coins, we can't compute that.

Wait, for example, if they had coins of different denominations, maybe they could have different leftovers. But the problem doesn't specify the number of coins or their denominations, so the only way to answer is to assume that all their money was in coins, which when converted into dollars, leaves the cents. Therefore, the answer is 59 cents.

Alternatively, perhaps there is some trick. For example, the total amount is $126.59. If they convert as many coins as possible into dollars, meaning they take all the coins and exchange them for bills. However, in reality, the coins are already part of the total amount. So, for example, if you have $126.59, which is 126 dollars and 59 cents. The 126 dollars is in bills, and the 59 cents is in coins. But if you want to convert the coins into dollars, you can't. Therefore, the leftover is 59 cents. Therefore, answer is 59.

But maybe the problem is in another way. For example, if they had all their money in coins, and they wanted to convert them into bills, but in the process, they might have a certain number of coins, and when exchanged, they get bills. However, since the problem does not give the number of coins or their denominations, we cannot compute the exact amount of leftover coins. Therefore, the only logical way is to assume that the total amount is in cents, so converting into dollars gives us total cents divided by 100, remainder is the leftover. Therefore, 12659 cents divided by 100 is 126 with remainder 59. Therefore, the answer is 59 cents. So I think that is the answer.

But to check, maybe there's a different approach. Let me think. Suppose that the question is about the coins they have left after converting as much as possible into bills. But if they have total amount of money, which is $126.59, and they want to convert coins into bills. If they have all the money in coins, then they can exchange all the cents (i.e., 59 cents) as coins into dollars? Wait, no, you can't do that. To get a dollar bill, you need 100 cents. So they can't convert 59 cents into a dollar bill. Therefore, they can't convert any of the coins into dollar bills. Therefore, the leftover coins are 59 cents. So same answer.

Alternatively, if they had more coins. For example, if they had more than $126.59 in coins, but no, the total is fixed. Wait, the problem says they pooled their money together, so they have that $126.59. So if they have this amount in coins, the answer is 59 cents.

Alternatively, if the question is trying to trick us into thinking that they have to convert all coins into dollars, which would mean that they have to break down the total into dollars and cents, and the leftover cents would be the remainder. Which again, is 59 cents. So, regardless of how I think about it, the answer is 59 cents.

Therefore, I think the answer is 59. But just to make sure, let me check my total cents again.

Original amounts:

Linda: $27.47 = 2747 cents

Sherry: $35.23 = 3523 cents

June: $37.37 = 3737 cents

Connie: $26.52 = 2652 cents

Adding them:

2747 + 3523. Let me add them again:

2747 + 3523:

First, 2000 + 3000 = 5000

700 + 500 = 1200, so 5000 + 1200 = 6200

47 + 23 = 70. So total is 6200 + 70 = 6270 cents. Correct.

Then 3737 + 2652:

3737 + 2652:

3000 + 2000 = 5000

700 + 600 = 1300, 5000 + 1300 = 6300

37 + 52 = 89. So total 6300 + 89 = 6389. Correct.

Then 6270 + 6389:

6270 + 6389:

Let me split 6389 into 6000 + 389.

6270 + 6000 = 12270.

Then 12270 + 389 = 12659 cents. Correct.

Divide by 100: 12659 / 100 = 126.59. So remainder is 59 cents. Therefore, yes, 59 cents.

Therefore, the answer is 59 cents. So the problem is straightforward once you convert all the money into cents. So I think that is correct.

**Final Answer**
\boxed{59}

To solve this problem, we need to determine how much **money in cents** was left after Linda, Sherry, June, and Connie **converted as many coins as possible into dollars**. The key here is that although the individuals earned amounts in **dollars**, the bank process involves **converting all money into bills** as much as possible, which implies we're dealing with **coins** and want to determine the **leftover coins (in cents)** after converting as much as possible into **whole dollars**.

---

### Step 1: Convert Each Earnings to Cents

We begin by converting each person's earnings from dollars and cents into total **cents**:

- **Linda**:
  $27.47 = 27 \times 100 + 47 = 2747$ cents
- **Sherry**:
  $35.23 = 35 \times 100 + 23 = 3523$ cents
- **June**:
  $37.37 = 37 \times 100 + 37 = 3737$ cents
- **Connie**:
  $26.52 = 26 \times 100 + 52 = 2652$ cents

---

### Step 2: Add All Cents Together

Now, let's compute the total number of cents they have:

$$
2747 + 3523 + 3737 + 2652 = 12659 \text{ cents}
$$

---

### Step 3: Convert as Many Cents as Possible into Dollars

Since 1 dollar = 100 cents, we can convert whole dollars by dividing the total cents by 100. The **remainder** of this division tells us how many **cents were left over** after converting as many as possible into dollars.

$$
12659 \div 100 = 126 \text{ dollars with remainder } 59 \text{ cents}
$$

---

### Final Result

After converting as many cents as possible into dollars, the **total leftover** is just the **remainder**, which is:

$$
\boxed{59}
$$

[Assistant]
Explanation:
Okay, so Linda, Sherry, June, and Connie sold cookies and earned different amounts of money. Now they want to pool their money together, convert coins into dollars, and figure out how much money in cents is left after they convert as many coins as possible into bills. Hmm, let me think about how to approach this.

First, I need to figure out the total amount of money they have. Let me write down each person's earnings:

- Linda: $27.47
- Sherry: $35.23
- June: $37.37
- Connie: $26.52

So, adding these up. Let me do that step by step. Let's start with Linda and Sherry. 27.47 plus 35.23. Let me add the dollars and cents separately. 27 + 35 is 62, and 47 cents + 23 cents is 70 cents. So that's $62.70 together.

Then June and Connie: 37.37 + 26.52. Again, dollars and cents. 37 + 26 is 63, and 37 + 52 is 89 cents. So that's $63.89.

Now adding those two totals together: $62.70 + $63.89. Let's do that. 62 + 63 is 125, and 70 cents + 89 cents is 159 cents. 159 cents is $1.59. So adding that to $125 gives $126.59 total. Wait, let me check that again. Because 62.70 + 63.89. Let me add the decimal parts first: 0.70 + 0.89 = 1.59. Then the whole dollars: 62 + 63 = 125. So total is 125 + 1.59 = $126.59. Yes, that seems right.

So total money they have is $126.59. Now, the problem says they pooled their money together and went to the bank to convert the coins into dollars. How much money in cents was left after they converted as many of the coins into bills as they could?

Wait, so they have a certain amount of money, which is $126.59. Now, they want to convert as many coins into bills as possible. So, this is about converting coins to bills, so the leftover would be the coins that couldn't be converted into whole dollars?

But wait, the question is a bit confusing. Wait, when you convert coins into bills, you can exchange coins for bills. But if you have, for example, a certain amount of coins, you can exchange them for bills. However, the problem says "how much money, in cents, was left after they converted as many of the coins into bills as they could."

Wait, hold on. Maybe the initial amounts they earned include both coins and bills? Wait, but the problem says they earned amounts in dollars, like $27.47, which is dollars and cents. But when they pooled their money together, maybe they have a total amount of money in dollars and cents, and then they wanted to convert all coins into bills. However, the problem says "converted the coins into dollars". So, perhaps they have coins, and they want to exchange coins for dollar bills. But how does that work?

Wait, maybe they have all their money as coins, and they want to convert them into dollars. But wait, the problem says they "pooled their money together and went to the bank to convert the coins into dollars." So perhaps the question is about converting the coins into bills, meaning exchanging coins for paper money, and they want to know how much is left in coins after converting as much as possible to bills. But if they have total money, which is $126.59, which is in dollars and cents, then to convert coins into dollars, they would take the coins and exchange them for dollar bills. But if they have, say, a certain amount of coins, maybe the total amount they have is in coins. But the original problem says they earned amounts in dollars, which could be in bills and coins. Wait, this is confusing.

Wait, maybe the problem is that they have all their money in coins, and they want to convert as many coins as possible into bills, so the leftover is the coins that cannot be converted into whole dollars. But the total amount is $126.59. But if they have that amount in coins, then converting into bills would be taking the coins and exchanging them for bills. But how?

Alternatively, maybe the problem is that they have a certain amount of cents, and they want to convert as many as possible into dollars, so the leftover would be the cents left after dividing by 100. Wait, but the total amount is in dollars. Wait, the original total is $126.59, which is 126 dollars and 59 cents. If they convert all the coins into bills, they would take the cents and convert them into dollars? No, that doesn't make sense. Wait, maybe they have all their money in coins, which they want to convert into dollars. But the problem says they "pooled their money together and went to the bank to convert the coins into dollars." So, maybe they have all their money in coins, and they want to convert them into dollar bills. But how would that work? For example, if you have coins, you can exchange them for bills. But the question is about the leftover coins after converting as much as possible. However, if they convert all the coins into bills, there would be no coins left. That seems conflicting.

Wait, maybe the problem is actually about the cents left after converting the total amount into dollars and cents. Wait, but the question says "convert the coins into dollars," so perhaps the total amount they have is in cents. Wait, but the original amounts were in dollars. Wait, maybe the problem is that when they pooled their money together, they had a total amount in cents, and they wanted to convert as much as possible into dollars (i.e., take cents and convert them into dollars), leaving some cents. But that doesn't make sense because converting cents into dollars would mean giving up coins for bills. Wait, but if they have the money in cents, then converting into dollars would be taking the total cents and converting into dollars, so the leftover would be the remaining cents. Wait, maybe the problem is that they have a total amount of money, which is in dollars and cents, and the question is asking about the cents left after converting the coins into dollars. Wait, but how do we know how much was in coins vs. bills?

Hold on, maybe the key here is that when you have money, it's made up of bills and coins. But the problem says they pooled their money together and went to the bank to convert the coins into dollars. So maybe they have some number of coins, and they want to exchange them for dollar bills. But if they have, say, $126.59 in total, which is 126 dollars and 59 cents. If they want to convert the coins into dollars, that implies that they have the 59 cents in coins, and they want to convert those into dollars. But you can't convert cents into dollars unless you have more than a dollar. Wait, this is confusing.

Alternatively, maybe the problem is that when they had the money, they had it in coins, and they wanted to convert them into bills. So, for example, if they have coins, they can exchange them for bills. But when you exchange coins into bills, you can exchange any number of coins into bills. But if they want to convert as many coins as possible into bills, then they would take all the coins and exchange them for the maximum number of bills, and the leftover would be the coins that couldn't be converted. But if they have, for example, 59 cents, which is less than a dollar, then they can't convert that into a dollar bill. Therefore, the leftover would be 59 cents. Wait, but that seems too straightforward. However, the total amount is $126.59. If they have all that as coins, then converting as much as possible into bills would be converting all the whole dollars into bills and leaving the 59 cents as coins. Therefore, the leftover would be 59 cents. But that answer seems too easy, and maybe I'm missing something here.

Wait, but the problem says they "converted the coins into dollars". So maybe they have some number of coins, and they try to convert them into dollar bills. But the problem is that you can't convert less than a dollar into a bill. Therefore, the leftover would be the cents that couldn't be converted into a dollar. However, if they have, for example, $126.59 total, that is 126 dollars and 59 cents. If they have the 126 dollars as bills and the 59 cents in coins, then converting the coins into dollars... you can't convert 59 cents into a dollar bill. Therefore, they can't convert any coins into dollars, so the leftover is 59 cents. But that would be the answer. But why would the problem mention all their earnings? Maybe the problem is not about converting the cents, but something else?

Wait, perhaps the problem is that when they pooled their money together, they had a total amount in cents, and they wanted to convert as many cents as possible into dollars, leaving some cents. But if that's the case, then we need to convert the total amount into cents, then divide by 100 to find how many dollars they can convert, and the remainder would be the leftover cents. But the problem says "convert the coins into dollars", which might imply that the total amount is in coins. But how do we know how much was in coins?

Wait, perhaps the problem is that when they earned money, they received it in coins, so each of their amounts is in cents, and they want to convert as many as possible into dollars. Wait, but the problem says they earned amounts in dollars, like $27.47, which is dollars and cents. So if we convert each amount into cents, we can get the total number of cents they had. Then, convert as many as possible into dollars (which would be dividing the total cents by 100) and the remainder is the leftover cents. So maybe that's the case.

Let me try this approach. So, first convert each person's earnings into cents:

Linda: $27.47 = 27*100 + 47 = 2700 + 47 = 2747 cents

Sherry: $35.23 = 35*100 + 23 = 3523 cents

June: $37.37 = 37*100 + 37 = 3737 cents

Connie: $26.52 = 26*100 + 52 = 2652 cents

Total cents = 2747 + 3523 + 3737 + 2652.

Let me compute that step by step:

First, 2747 + 3523. Let's add 2747 + 3523. 2700 + 3500 = 6200. 47 + 23 = 70. So total is 6200 + 70 = 6270 cents.

Then, 3737 + 2652. Let's compute that. 3700 + 2600 = 6300. 37 + 52 = 89. So total is 6300 + 89 = 6389 cents.

Now, total cents is 6270 + 6389. Let's compute that.

6270 + 6389. Let me add 6270 + 6000 = 12270, then add 389: 12270 + 389 = 12659 cents.

So total cents is 12,659 cents. So if they converted as many as possible into dollars, how many dollars is that? Well, dividing 12659 by 100 gives 126 dollars with a remainder of 59 cents. Therefore, the leftover is 59 cents. Therefore, the answer is 59 cents. So, $\boxed{59}$.

Wait, but this is the same answer as before. So whether you think of the total amount as $126.59, which is 126 dollars and 59 cents, and thus leftover is 59 cents. But why did they mention all the individual amounts? Because if you just added up the total, you get $126.59, which is equivalent to 12659 cents, so converting into dollars gives 126 dollars and 59 cents, so leftover is 59 cents. But maybe there is a trick here?

Wait, but is there a possibility that they have coins and bills, and converting coins into dollars would involve exchanging coins for bills, but if they have some coins, they can exchange them. Wait, but if they have $126.59 in total, and they have some combination of bills and coins. For example, they may have a certain number of coins, and they want to exchange those coins into bills. However, unless we know how many coins they have, we can't determine the leftover. But the problem says "converted the coins into dollars." So, perhaps the question is assuming that all the money is in coins. Therefore, converting as much as possible into dollars would mean taking all the coins and exchanging them for dollars, which would leave you with the leftover cents. Therefore, since total cents is 12659, the leftover is 59 cents. Therefore, answer is 59.

Alternatively, maybe the question is about converting the coins into dollar bills, meaning exchanging coins for dollar bills, but if you have, for example, 126 dollars and 59 cents, the 126 dollars is already in bills, so you can't convert that into more bills. Whereas the 59 cents is in coins, so you can't convert that into a bill. Therefore, leftover is 59 cents. So regardless of interpretation, the answer seems to be 59 cents.

But why would the problem give all the individual amounts? Because if you just take the total amount, $126.59, and convert into cents, you get 12659 cents. Dividing by 100, the remainder is 59. Therefore, 59 cents left. But maybe there's another interpretation. Maybe they had coins, and when they went to the bank, they wanted to exchange coins for bills, but they had different denominations. Wait, but unless we know the exact number of coins, we can't compute that.

Wait, for example, if they had coins of different denominations, maybe they could have different leftovers. But the problem doesn't specify the number of coins or their denominations, so the only way to answer is to assume that all their money was in coins, which when converted into dollars, leaves the cents. Therefore, the answer is 59 cents.

Alternatively, perhaps there is some trick. For example, the total amount is $126.59. If they convert as many coins as possible into dollars, meaning they take all the coins and exchange them for bills. However, in reality, the coins are already part of the total amount. So, for example, if you have $126.59, which is 126 dollars and 59 cents. The 126 dollars is in bills, and the 59 cents is in coins. But if you want to convert the coins into dollars, you can't. Therefore, the leftover is 59 cents. Therefore, answer is 59.

But maybe the problem is in another way. For example, if they had all their money in coins, and they wanted to convert them into bills, but in the process, they might have a certain number of coins, and when exchanged, they get bills. However, since the problem does not give the number of coins or their denominations, we cannot compute the exact amount of leftover coins. Therefore, the only logical way is to assume that the total amount is in cents, so converting into dollars gives us total cents divided by 100, remainder is the leftover. Therefore, 12659 cents divided by 100 is 126 with remainder 59. Therefore, the answer is 59 cents. So I think that is the answer.

But to check, maybe there's a different approach. Let me think. Suppose that the question is about the coins they have left after converting as much as possible into bills. But if they have total amount of money, which is $126.59, and they want to convert coins into bills. If they have all the money in coins, then they can exchange all the cents (i.e., 59 cents) as coins into dollars? Wait, no, you can't do that. To get a dollar bill, you need 100 cents. So they can't convert 59 cents into a dollar bill. Therefore, they can't convert any of the coins into dollar bills. Therefore, the leftover coins are 59 cents. So same answer.

Alternatively, if they had more coins. For example, if they had more than $126.59 in coins, but no, the total is fixed. Wait, the problem says they pooled their money together, so they have that $126.59. So if they have this amount in coins, the answer is 59 cents.

Alternatively, if the question is trying to trick us into thinking that they have to convert all coins into dollars, which would mean that they have to break down the total into dollars and cents, and the leftover cents would be the remainder. Which again, is 59 cents. So, regardless of how I think about it, the answer is 59 cents.

Therefore, I think the answer is 59. But just to make sure, let me check my total cents again.

Original amounts:

Linda: $27.47 = 2747 cents

Sherry: $35.23 = 3523 cents

June: $37.37 = 3737 cents

Connie: $26.52 = 2652 cents

Adding them:

2747 + 3523. Let me add them again:

2747 + 3523:

First, 2000 + 3000 = 5000

700 + 500 = 1200, so 5000 + 1200 = 6200

47 + 23 = 70. So total is 6200 + 70 = 6270 cents. Correct.

Then 3737 + 2652:

3737 + 2652:

3000 + 2000 = 5000

700 + 600 = 1300, 5000 + 1300 = 6300

37 + 52 = 89. So total 6300 + 89 = 6389. Correct.

Then 6270 + 6389:

6270 + 6389:

Let me split 6389 into 6000 + 389.

6270 + 6000 = 12270.

Then 12270 + 389 = 12659 cents. Correct.

Divide by 100: 12659 / 100 = 126.59. So remainder is 59 cents. Therefore, yes, 59 cents.

Therefore, the answer is 59 cents. So the problem is straightforward once you convert all the money into cents. So I think that is correct.

**Final Answer**
\boxed{59}

To solve this problem, we need to determine how much **money in cents** was left after Linda, Sherry, June, and Connie **converted as many coins as possible into dollars**. The key here is that although the individuals earned amounts in **dollars**, the bank process involves **converting all money into bills** as much as possible, which implies we're dealing with **coins** and want to determine the **leftover coins (in cents)** after converting as much as possible into **whole dollars**.

---

### Step 1: Convert Each Earnings to Cents

We begin by converting each person's earnings from dollars and cents into total **cents**:

- **Linda**:
  $27.47 = 27 \times 100 + 47 = 2747$ cents
- **Sherry**:
  $35.23 = 35 \times 100 + 23 = 3523$ cents
- **June**:
  $37.37 = 37 \times 100 + 37 = 3737$ cents
- **Connie**:
  $26.52 = 26 \times 100 + 52 = 2652$ cents

---

### Step 2: Add All Cents Together

Now, let's compute the total number of cents they have:

$$
2747 + 3523 + 3737 + 2652 = 12659 \text{ cents}
$$

---

### Step 3: Convert as Many Cents as Possible into Dollars

Since 1 dollar = 100 cents, we can convert whole dollars by dividing the total cents by 100. The **remainder** of this division tells us how many **cents were left over** after converting as many as possible into dollars.

$$
12659 \div 100 = 126 \text{ dollars with remainder } 59 \text{ cents}
$$

---

### Final Result

After converting as many cents as possible into dollars, the **total leftover** is just the **remainder**, which is:

$$
\boxed{59}
$$

Solution:
Instead of adding large numbers together, we can find the residue for each person for easier computation. We convert the amount they earned to cents and find the modulo $100$ for each.  \begin{align*}
2747 &\equiv 47 \pmod{100},\\
3523 &\equiv 23 \pmod{100},\\
3737 &\equiv 37 \pmod{100},\\
2652 &\equiv 52 \pmod{100}
\end{align*}We want to find the modulo $100$ of the total number of cents. We can add the separate residues to get $$47+23+37+52 \equiv 159 \equiv 59 \pmod{100}$$Therefore, they have $\boxed{59}$ cents left after converting as much of the money into bills as possible.


Explanation:
Okay, so I need to figure out for how many values of x the expression (x - 5)/(x² - 9) is not defined. Hmm, let me recall. When is a rational expression not defined? Well, it's undefined when the denominator is zero because you can't divide by zero. So my job is to find all the x values that make the denominator zero.

The denominator here is x² - 9. So I need to solve the equation x² - 9 = 0. Let me do that.

x² - 9 = 0. This looks like a difference of squares, right? Because x² is a square and 9 is 3 squared. So, the difference of squares formula is a² - b² = (a - b)(a + b). Applying that here, I can factor the denominator:

x² - 9 = (x - 3)(x + 3).

So setting that equal to zero:

(x - 3)(x + 3) = 0.

To find the solutions, I can use the zero product property, which says that if the product of two factors is zero, then at least one of the factors must be zero. So:

x - 3 = 0 or x + 3 = 0.

Solving those:

For x - 3 = 0, adding 3 to both sides gives x = 3.

For x + 3 = 0, subtracting 3 from both sides gives x = -3.

So the denominator is zero when x is 3 or -3. Therefore, the expression (x - 5)/(x² - 9) is not defined at x = 3 and x = -3.

Wait, let me make sure I didn't miss anything. The numerator is x - 5, but that doesn't affect where the expression is undefined. Even if the numerator was zero, the expression would still be undefined if the denominator is zero. So, the only values that matter are the ones that make the denominator zero.

Therefore, there are two values of x where the expression is undefined: -3 and 3.

But let me check if there's any other possible way this could be undefined. For example, could there be any restrictions from the numerator? Let me think. If the numerator was something else, but no, even if the numerator was zero, the expression would still be defined (as long as denominator isn't zero). If both numerator and denominator are zero, that's an indeterminate form, but the expression itself would still be undefined at that point. However, in this case, the numerator is x - 5. Let me check if x = 3 or x = -3 makes the numerator zero.

For x = 3: numerator is 3 - 5 = -2, which is not zero.

For x = -3: numerator is -3 - 5 = -8, which is also not zero.

Therefore, at both x = 3 and x = -3, the denominator is zero and the numerator is not zero, so the expression is undefined at those points.

Therefore, there are two values where the expression is undefined. So the answer should be 2.

But wait, let me check if there are more values. Suppose the denominator had more factors or something. Let me check the denominator again: x² - 9. That factors into (x - 3)(x + 3), so quadratic, degree 2, so it can have at most two real roots. Which we found: 3 and -3. So there are exactly two values where denominator is zero. Hence, the expression is undefined at exactly two values of x.

Therefore, the answer is 2.

But just to be thorough, maybe I can plug in those values into the original expression to check. Let me try x = 3:

Plug x = 3 into denominator: 3² - 9 = 9 - 9 = 0. So denominator is zero. Therefore, expression is undefined. Similarly, x = -3: (-3)^2 - 9 = 9 - 9 = 0, same thing.

Therefore, that's two values.

Is there any other x that could make denominator zero? Let me solve x² - 9 = 0 again.

x² = 9, so x = sqrt(9) or x = -sqrt(9), which is 3 or -3. So, that's it.

Therefore, the answer is two values.

Wait, but the question says "for how many values of x is the expression... not defined?" So it's asking for the number of x-values that make the expression undefined, which as we found are two. So the answer is 2.

But just to check if there's a trick here. For example, sometimes people might think if there's a common factor between numerator and denominator, but in this case, the numerator is x - 5 and denominator is x² - 9, which factors into (x - 3)(x + 3). There's no common factors here. So even if there was a common factor, like if the numerator was x - 3, then we would have a common factor, but in this case, no. Therefore, the expression is undefined only at x = 3 and x = -3.

So yeah, I think that's solid. Therefore, the answer is 2.

**Final Answer**
The expression is not defined for \boxed{2} values of $ x $.

To determine for how many values of $ x $ the expression

$$
\frac{x - 5}{x^2 - 9}
$$

is **not defined**, we must identify the points where the expression becomes undefined. A rational expression like this is undefined when the **denominator is zero**, since division by zero is not defined in real numbers.

---

### Step 1: Identify the Denominator

The denominator of the expression is $ x^2 - 9 $. We set this equal to zero and solve:

$$
x^2 - 9 = 0
$$

This equation is a **difference of squares**, so we can factor it as:

$$
x^2 - 9 = (x - 3)(x + 3)
$$

Setting each factor equal to zero:

$$
x - 3 = 0 \quad \Rightarrow \quad x = 3
$$
$$
x + 3 = 0 \quad \Rightarrow \quad x = -3
$$

So the denominator is zero when $ x = 3 $ or $ x = -3 $.

---

### Step 2: Check if Numerator is Zero at These Points

Even though the expression becomes undefined at these points, it’s useful to check if the numerator is also zero. If both the numerator and denominator are zero, the expression would be in a form of an indeterminate form (like $ \frac{0}{0} $), but in that case, the function could still be defined through limits or simplification. However, for the **purpose of determining where the expression is undefined**, we only care about the **denominator**.

Let's evaluate the numerator $ x - 5 $ at the two values:

- When $ x = 3 $, the numerator is $ 3 - 5 = -2 $
- When $ x = -3 $, the numerator is $ -3 - 5 = -8 $

In both cases, the numerator is **not zero**, so the expression is indeed **undefined** at those points.

---

### Step 3: Conclusion

The expression $ \frac{x - 5}{x^2 - 9} $ is not defined **only** when the denominator is zero. This occurs at **two distinct** values of $ x $: $ x = 3 $ and $ x = -3 $.

There are **no other real values** that make the denominator zero. Since it's a quadratic equation, there are at most two real roots, and we've found both of them.

---

### Final Answer

$$
\boxed{2}
$$

[Assistant]
Explanation:
Okay, so I need to figure out for how many values of x the expression (x - 5)/(x² - 9) is not defined. Hmm, let me recall. When is a rational expression not defined? Well, it's undefined when the denominator is zero because you can't divide by zero. So my job is to find all the x values that make the denominator zero.

The denominator here is x² - 9. So I need to solve the equation x² - 9 = 0. Let me do that.

x² - 9 = 0. This looks like a difference of squares, right? Because x² is a square and 9 is 3 squared. So, the difference of squares formula is a² - b² = (a - b)(a + b). Applying that here, I can factor the denominator:

x² - 9 = (x - 3)(x + 3).

So setting that equal to zero:

(x - 3)(x + 3) = 0.

To find the solutions, I can use the zero product property, which says that if the product of two factors is zero, then at least one of the factors must be zero. So:

x - 3 = 0 or x + 3 = 0.

Solving those:

For x - 3 = 0, adding 3 to both sides gives x = 3.

For x + 3 = 0, subtracting 3 from both sides gives x = -3.

So the denominator is zero when x is 3 or -3. Therefore, the expression (x - 5)/(x² - 9) is not defined at x = 3 and x = -3.

Wait, let me make sure I didn't miss anything. The numerator is x - 5, but that doesn't affect where the expression is undefined. Even if the numerator was zero, the expression would still be undefined if the denominator is zero. So, the only values that matter are the ones that make the denominator zero.

Therefore, there are two values of x where the expression is undefined: -3 and 3.

But let me check if there's any other possible way this could be undefined. For example, could there be any restrictions from the numerator? Let me think. If the numerator was something else, but no, even if the numerator was zero, the expression would still be defined (as long as denominator isn't zero). If both numerator and denominator are zero, that's an indeterminate form, but the expression itself would still be undefined at that point. However, in this case, the numerator is x - 5. Let me check if x = 3 or x = -3 makes the numerator zero.

For x = 3: numerator is 3 - 5 = -2, which is not zero.

For x = -3: numerator is -3 - 5 = -8, which is also not zero.

Therefore, at both x = 3 and x = -3, the denominator is zero and the numerator is not zero, so the expression is undefined at those points.

Therefore, there are two values where the expression is undefined. So the answer should be 2.

But wait, let me check if there are more values. Suppose the denominator had more factors or something. Let me check the denominator again: x² - 9. That factors into (x - 3)(x + 3), so quadratic, degree 2, so it can have at most two real roots. Which we found: 3 and -3. So there are exactly two values where denominator is zero. Hence, the expression is undefined at exactly two values of x.

Therefore, the answer is 2.

But just to be thorough, maybe I can plug in those values into the original expression to check. Let me try x = 3:

Plug x = 3 into denominator: 3² - 9 = 9 - 9 = 0. So denominator is zero. Therefore, expression is undefined. Similarly, x = -3: (-3)^2 - 9 = 9 - 9 = 0, same thing.

Therefore, that's two values.

Is there any other x that could make denominator zero? Let me solve x² - 9 = 0 again.

x² = 9, so x = sqrt(9) or x = -sqrt(9), which is 3 or -3. So, that's it.

Therefore, the answer is two values.

Wait, but the question says "for how many values of x is the expression... not defined?" So it's asking for the number of x-values that make the expression undefined, which as we found are two. So the answer is 2.

But just to check if there's a trick here. For example, sometimes people might think if there's a common factor between numerator and denominator, but in this case, the numerator is x - 5 and denominator is x² - 9, which factors into (x - 3)(x + 3). There's no common factors here. So even if there was a common factor, like if the numerator was x - 3, then we would have a common factor, but in this case, no. Therefore, the expression is undefined only at x = 3 and x = -3.

So yeah, I think that's solid. Therefore, the answer is 2.

**Final Answer**
The expression is not defined for \boxed{2} values of $ x $.

To determine for how many values of $ x $ the expression

$$
\frac{x - 5}{x^2 - 9}
$$

is **not defined**, we must identify the points where the expression becomes undefined. A rational expression like this is undefined when the **denominator is zero**, since division by zero is not defined in real numbers.

---

### Step 1: Identify the Denominator

The denominator of the expression is $ x^2 - 9 $. We set this equal to zero and solve:

$$
x^2 - 9 = 0
$$

This equation is a **difference of squares**, so we can factor it as:

$$
x^2 - 9 = (x - 3)(x + 3)
$$

Setting each factor equal to zero:

$$
x - 3 = 0 \quad \Rightarrow \quad x = 3
$$
$$
x + 3 = 0 \quad \Rightarrow \quad x = -3
$$

So the denominator is zero when $ x = 3 $ or $ x = -3 $.

---

### Step 2: Check if Numerator is Zero at These Points

Even though the expression becomes undefined at these points, it’s useful to check if the numerator is also zero. If both the numerator and denominator are zero, the expression would be in a form of an indeterminate form (like $ \frac{0}{0} $), but in that case, the function could still be defined through limits or simplification. However, for the **purpose of determining where the expression is undefined**, we only care about the **denominator**.

Let's evaluate the numerator $ x - 5 $ at the two values:

- When $ x = 3 $, the numerator is $ 3 - 5 = -2 $
- When $ x = -3 $, the numerator is $ -3 - 5 = -8 $

In both cases, the numerator is **not zero**, so the expression is indeed **undefined** at those points.

---

### Step 3: Conclusion

The expression $ \frac{x - 5}{x^2 - 9} $ is not defined **only** when the denominator is zero. This occurs at **two distinct** values of $ x $: $ x = 3 $ and $ x = -3 $.

There are **no other real values** that make the denominator zero. Since it's a quadratic equation, there are at most two real roots, and we've found both of them.

---

### Final Answer

$$
\boxed{2}
$$

Solution: The expression is not defined when the denominator is equal to zero. Thus, we need to find the number of values of $x$ such that $x^2-9=0$. Rearranging the equation and taking the square root of both sides, we have $x^2-9=0\Rightarrow x^2=9 \Rightarrow x=\pm3$. Thus, there are $\boxed{2}$ different values of $x$ that make the given expression undefined.


Explanation:
Okay, so I need to find the area of the shaded triangle in the diagram. Let me try to visualize the Asymptote code they provided. It seems like there's a rectangle with length 10 cm and height 3 cm. The rectangle is drawn from (0,0) to (10,0) to (10,3) to (0,3) and back. Then there are some lines drawn: from (0,0) to (2,3) to (10,0). The shaded triangle is formed by connecting these three points: (0,0), (2,3), and (10,0).

First, let me recall the formula for the area of a triangle. The formula is (base * height)/2. But I need to figure out the base and the height of the shaded triangle. Alternatively, since I have coordinates for all three vertices, maybe I can use the coordinate geometry formula for area.

Let me think. If I have three points, say (x1,y1), (x2,y2), (x3,y3), the area can be calculated using the determinant formula:

Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)) / 2|

Alternatively, I can use vectors or base and height. Maybe I can try both methods to check.

First, let me try coordinates. The three vertices of the triangle are (0,0), (2,3), and (10,0). Let me label them as A(0,0), B(2,3), and C(10,0).

Using the determinant formula:

Area = |(Ax(By - Cy) + Bx(Cy - Ay) + Cx(Ay - By))/2|

Plugging in:

Ax = 0, Ay = 0

Bx = 2, By = 3

Cx = 10, Cy = 0

So,

Area = |0*(3 - 0) + 2*(0 - 0) + 10*(0 - 3)| / 2

Calculating step by step:

First term: 0*(3 - 0) = 0*3 = 0

Second term: 2*(0 - 0) = 2*0 = 0

Third term: 10*(0 - 3) = 10*(-3) = -30

Adding them up: 0 + 0 + (-30) = -30

Take absolute value and divide by 2: | -30 | / 2 = 30 / 2 = 15

So, the area is 15 cm². Hmm, that seems straightforward. But wait, let me verify with another method to ensure I didn't make a mistake.

Alternatively, since the triangle has vertices at (0,0), (2,3), and (10,0), maybe I can split it into simpler shapes or use base and height.

Looking at the coordinates, points A(0,0) and C(10,0) are on the x-axis, so the base AC is 10 cm. But the triangle isn't necessarily with base AC and height from B to AC. Wait, point B is at (2,3). If I consider AC as the base, which is 10 cm, then the height would be the vertical distance from point B to the base AC. However, since AC is along the x-axis, the y-coordinate of B is 3 cm. Wait, but that would make the area (base * height)/2 = (10 * 3)/2 = 15 cm². That gives the same result as the determinant method! So that's reassuring.

But hold on, is that correct? Because the triangle is not necessarily a right triangle. Wait, point B is at (2,3), so if I drop a perpendicular from B to AC (which is the x-axis), the height is indeed 3 cm, since the y-coordinate is 3. So the height is 3 cm. So area is (10*3)/2 = 15 cm². So both methods give the same result.

But wait, but in the diagram, there is a rectangle with sides 10 cm and 3 cm. The shaded triangle is inside this rectangle. If the triangle has area 15 cm², then since the rectangle is 10*3=30 cm², the triangle would be exactly half of the rectangle. But is that true? Because the triangle connects (0,0), (2,3), (10,0). Wait, but if you connect (0,0) to (2,3) to (10,0), then the triangle is not the entire half of the rectangle. Wait, but if you connect (0,0) to (10,0) to (0,3) to (0,0), that's a triangle with area (10*3)/2 = 15 cm². But the triangle in question is different. Wait, but according to our calculation, it's also 15 cm². So maybe it's equivalent? Wait, let me think.

Wait, let me visualize again. The rectangle is from (0,0) to (10,0) to (10,3) to (0,3). So the rectangle is 10 units long and 3 units high. The shaded triangle is formed by connecting (0,0), (2,3), and (10,0). So imagine the triangle that starts at the bottom left corner, goes up to a point 2 units along the top edge (since the top edge is from (0,3) to (10,3), and the point is at (2,3)), then goes down to the bottom right corner. So this triangle is not the same as the triangle formed by (0,0), (0,3), (10,0), which would be a triangle with area 15 cm². But according to our calculation, the triangle (0,0), (2,3), (10,0) also has area 15 cm². That seems counterintuitive. How can two different triangles have the same area? Well, maybe they do. Let me check with another approach.

Alternatively, maybe we can use coordinates to calculate the area. Let's consider coordinates again. If I split the triangle into parts or use coordinates. Another way to calculate the area is using the shoelace formula, which is the same as the determinant formula.

Shoelace formula:

List the coordinates in order, repeating the first at the end:

(0,0), (2,3), (10,0), (0,0)

Compute sum of products downward:

(0*3) + (2*0) + (10*0) = 0 + 0 + 0 = 0

Compute sum of products upward:

(0*2) + (3*10) + (0*0) = 0 + 30 + 0 = 30

Area = |0 - 30| / 2 = 30 / 2 = 15 cm². Same result.

Hmm. So according to this, the area is 15 cm². But let me think again. Maybe the confusion comes from the rectangle. If we think about the triangle from (0,0), (10,0), (0,3), that's 15 cm², but the triangle in the problem is a different triangle. However, the area is the same. Is that possible?

Alternatively, maybe there is a mistake in my reasoning. Let me think. If I imagine the line from (0,0) to (2,3), then from (2,3) to (10,0). So this triangle is cutting off a part of the rectangle. Let me try to compute the area by subtracting some areas from the rectangle.

Alternatively, maybe using coordinates again. Let me try to find the equations of the lines and see.

First, the line from (0,0) to (2,3). The slope is (3 - 0)/(2 - 0) = 3/2. So the equation is y = (3/2)x.

Then, the line from (2,3) to (10,0). The slope is (0 - 3)/(10 - 2) = (-3)/8. So the equation is y - 3 = (-3/8)(x - 2). So y = (-3/8)x + (3/8)*2 + 3 = (-3/8)x + 6/8 + 24/8 = (-3/8)x + 30/8 = (-3/8)x + 15/4.

Now, if I want to find the area of the triangle, I can integrate or something, but that seems overkill. Alternatively, maybe find the intersection points or something else.

Alternatively, since we already have two methods giving 15 cm², maybe that's the answer. Wait, but why does the triangle have the same area as half the rectangle? Maybe the line from (0,0) to (2,3) divides the rectangle in some way?

Wait, but the rectangle has area 30 cm². If the triangle has area 15 cm², it's exactly half of the rectangle. Is there a reason for that? Maybe. For example, if you connect (0,0) to (2,3) to (10,0), maybe this triangle is congruent to another triangle in the rectangle? Or maybe it's symmetrical?

Alternatively, maybe not. Wait, let me think of coordinates. Let me try to use coordinate geometry to find the area between the lines. Wait, perhaps decomposing the figure.

Alternatively, if I consider the triangle ABC with A(0,0), B(2,3), C(10,0). The base AC is 10 cm, as before. The height is not 3 cm, because the height is the perpendicular distance from B to AC. Wait, earlier I thought that since AC is on the x-axis, the y-coordinate of B is 3, so that's the height. But is that correct?

Wait, if AC is along the x-axis from (0,0) to (10,0), then the line AC is y = 0. The height from point B(2,3) to AC is indeed the vertical distance from B to the line AC, which is 3 units. Therefore, the height is 3 cm. Therefore, area is (10*3)/2 = 15 cm². That seems correct.

But then why does the triangle (0,0), (2,3), (10,0) have the same area as (0,0), (0,3), (10,0)? Maybe because they are both triangles with the same base and same height? Wait, no. Wait, triangle (0,0), (0,3), (10,0) has base AC = 10 cm and height 3 cm. Triangle (0,0), (2,3), (10,0) also has base AC = 10 cm and height 3 cm? How is that possible? Wait, if the height is measured perpendicularly from the base AC, then regardless of where the third point is, as long as its vertical distance from AC is 3 cm, then the area would be same. But in this case, point (2,3) is vertically 3 cm above AC, yes. So even though it's not directly above the midpoint, the vertical distance is still 3 cm. Therefore, area is (base * height)/2. Hence 15 cm².

Therefore, maybe that is correct. However, let me try to think of another way to confirm.

Alternatively, perhaps using vectors. The vectors from A to B and from A to C.

Vector AB = (2,3) - (0,0) = (2,3)

Vector AC = (10,0) - (0,0) = (10,0)

The area of the triangle is half the magnitude of the cross product of AB and AC.

Cross product in 2D is scalar magnitude |Ax*By - Ay*Bx|. Wait, more accurately, for vectors AB and AC, the cross product is AB_x * AC_y - AB_y * AC_x.

So AB is (2,3), AC is (10,0). So cross product is 2*0 - 3*10 = 0 - 30 = -30. The magnitude is 30, so area is 30/2 = 15 cm². Yep, same answer again.

So all methods point to 15 cm². Therefore, maybe 15 is the correct answer.

But wait, the Asymptote code labels the rectangle with "10 cm" at the top and bottom, and "3 cm" on the left and right sides. So the rectangle is 10 cm by 3 cm. The triangle is inside this rectangle. The question is, is the area 15 cm²? That would be half of the rectangle. But let me think again. Suppose I have a rectangle of area 30 cm², and the triangle is half of that. Is that possible?

Yes, for example, if you draw a diagonal of the rectangle, you get two triangles each with area 15 cm². However, in this case, the triangle isn't a diagonal. It's connecting (0,0) to (2,3) to (10,0). But since the vertical distance from (2,3) to the base is 3 cm, same as the height of the rectangle, then the area would be same as the triangle formed by the diagonal. Wait, but the diagonal would be from (0,0) to (10,3), which is a different point. So why is the area same?

Wait, maybe because the average of the heights? Wait, not sure. But according to the calculations, yes.

Alternatively, maybe there's a misunderstanding. Let me check coordinates again.

If I imagine the rectangle, from (0,0) to (10,0) to (10,3) to (0,3). The shaded triangle is connecting (0,0) to (2,3) to (10,0). So this triangle is actually occupying the same area as the triangle connecting (0,0) to (10,0) to (0,3). Because both have base 10 cm and height 3 cm.

But how? Because the third point in the first triangle is (2,3), which is on the top edge of the rectangle. So maybe the area is same because regardless of where you put the third point on the top edge, if you connect it to (0,0) and (10,0), the area remains same?

Wait, let me test with another point. Suppose the third point is (5,3). Then the triangle would be (0,0), (5,3), (10,0). What is its area? Using the determinant formula:

Ax=0, Ay=0

Bx=5, By=3

Cx=10, Cy=0

Area = |0*(3 - 0) + 5*(0 - 0) + 10*(0 - 3)| / 2 = |0 + 0 + (-30)| / 2 = 30/2 = 15 cm². Same result.

So if you pick any point on the top edge of the rectangle (from (0,3) to (10,3)), connecting it to (0,0) and (10,0) gives a triangle of area 15 cm². That's interesting. So why is that?

Because the area of such a triangle is (base * height)/2, where base is 10 cm, and the height is 3 cm. Since the height is the vertical distance from the top edge to the base, which is 3 cm. So regardless of where the point is on the top edge, the height remains 3 cm. Therefore, the area remains same.

Therefore, even though the triangle is not a diagonal, it's similar in area.

So, in our problem, the third point is at (2,3), so it's 2 units from the left on the top edge, but the area is still 15 cm². Therefore, the answer is 15 cm².

But let me check with coordinates once more, just to make sure. Maybe drawing a diagram mentally. The triangle has vertices at (0,0), (2,3), and (10,0). If I imagine this triangle, it will have a base from (0,0) to (10,0), which is 10 cm. The third vertex is somewhere above the base, at (2,3). The height here is 3 cm as before. So, yes, the area is 1/2 * 10 * 3 = 15 cm².

Alternatively, I can think of the triangle as part of the rectangle. If I imagine the rectangle divided by the line from (0,0) to (2,3) and then from (2,3) to (10,0). Maybe we can find some other way to calculate the area. For example, breaking the rectangle into different regions.

Let me try that. The rectangle is 10 cm by 3 cm. Let's think of the triangle as being composed of two regions or something.

First, from (0,0) to (2,3) to (10,0). Let me see. Alternatively, perhaps compute the area of the triangle by integrating or using coordinates.

Alternatively, since I have two different methods giving me the same answer, and my reasoning checks out, maybe 15 cm² is correct.

But wait, let me think again. Let me check using base and height. Suppose I take another base. For instance, take the base as the segment from (0,0) to (2,3). Then I need to find the corresponding height.

But that might be more complicated. Let me try. The length of AB is sqrt((2-0)^2 + (3-0)^2) = sqrt(4 + 9) = sqrt(13). Then, to find the height corresponding to this base, I need the height from point C(10,0) to the line AB.

First, find the equation of line AB. As earlier, it's y = (3/2)x.

The distance from point C(10,0) to this line is |(3/2)(10) - 0 + 0| / sqrt((3/2)^2 + (-1)^2) ?

Wait, general formula for distance from point (x0, y0) to line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a² + b²). So let me rewrite the equation of AB in ax + by + c = 0 form.

Original line AB: y = (3/2)x => (3/2)x - y = 0. Multiply both sides by 2 to eliminate fraction: 3x - 2y = 0. So a = 3, b = -2, c = 0.

Distance from (10, 0):

|3*10 + (-2)*0 + 0| / sqrt(3² + (-2)^2) = |30| / sqrt(9 + 4) = 30 / sqrt(13).

Therefore, area of triangle is (base * height)/2 = (sqrt(13) * 30 / sqrt(13))/2 = 30/2 = 15 cm².

Same result again. Therefore, regardless of which base I choose, the area is 15 cm².

Therefore, I think I can be confident that the area is 15 cm².

But just to make sure, let me think of another approach. Maybe coordinate geometry with vectors.

The area can also be calculated by using coordinates and determinants. As done earlier, which gives 15.

Alternatively, maybe using matrices.

But I think all approaches confirm that the area is 15 cm².

Therefore, I think the answer is 15 cm².

**Final Answer**
The area of the shaded triangle is \boxed{15} square centimeters.

To determine the area of the shaded triangle shown in the diagram, we are provided with coordinates of its vertices:

- $ A = (0, 0) $
- $ B = (2, 3) $
- $ C = (10, 0) $

We are given that the rectangle has a base of 10 cm and height of 3 cm, and the shaded triangle lies entirely within this rectangle. The triangle is formed by connecting the points $ A $, $ B $, and $ C $.

---

### Step 1: Use the Coordinate Determinant Formula

The formula for the area of a triangle given its vertices $ (x_1, y_1) $, $ (x_2, y_2) $, and $ (x_3, y_3) $ is:

$$
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
$$

Substituting the coordinates of the triangle:

$$
\text{Area} = \frac{1}{2} \left| 0(3 - 0) + 2(0 - 0) + 10(0 - 3) \right|
$$
$$
= \frac{1}{2} \left| 0 + 0 - 30 \right| = \frac{1}{2} \times 30 = 15
$$

So the area of the triangle is **15 cm²**.

---

### Step 2: Reconfirm Using Base and Height

Let's consider the base of the triangle as the segment from $ A = (0, 0) $ to $ C = (10, 0) $. This forms a horizontal base of length 10 cm.

The height corresponds to the vertical distance from the third vertex $ B = (2, 3) $ to this base. Since the base lies along the x-axis, the y-coordinate of point $ B $ gives the height, which is:

$$
\text{Height} = 3 \text{ cm}
$$

Now, apply the area formula using base and height:

$$
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 3 = 15 \text{ cm}^2
$$

---

### Step 3: Geometric Insight (Optional)

The diagram shows that the triangle lies within a rectangle of area $ 10 \times 3 = 30 $ cm². If the triangle uses the same vertical height (3 cm) and base (10 cm), and its third vertex lies on the top edge of the rectangle, then the triangle's area is **exactly half the area of the rectangle**—regardless of where the third vertex is placed on the top edge, as long as the height remains 3 cm.

This provides an intuitive way to understand why the triangle’s area is consistent at 15 cm².

---

### Final Result

All approaches confirm the same result.

$$
\boxed{15}
$$

[Assistant]
Explanation:
Okay, so I need to find the area of the shaded triangle in the diagram. Let me try to visualize the Asymptote code they provided. It seems like there's a rectangle with length 10 cm and height 3 cm. The rectangle is drawn from (0,0) to (10,0) to (10,3) to (0,3) and back. Then there are some lines drawn: from (0,0) to (2,3) to (10,0). The shaded triangle is formed by connecting these three points: (0,0), (2,3), and (10,0).

First, let me recall the formula for the area of a triangle. The formula is (base * height)/2. But I need to figure out the base and the height of the shaded triangle. Alternatively, since I have coordinates for all three vertices, maybe I can use the coordinate geometry formula for area.

Let me think. If I have three points, say (x1,y1), (x2,y2), (x3,y3), the area can be calculated using the determinant formula:

Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)) / 2|

Alternatively, I can use vectors or base and height. Maybe I can try both methods to check.

First, let me try coordinates. The three vertices of the triangle are (0,0), (2,3), and (10,0). Let me label them as A(0,0), B(2,3), and C(10,0).

Using the determinant formula:

Area = |(Ax(By - Cy) + Bx(Cy - Ay) + Cx(Ay - By))/2|

Plugging in:

Ax = 0, Ay = 0

Bx = 2, By = 3

Cx = 10, Cy = 0

So,

Area = |0*(3 - 0) + 2*(0 - 0) + 10*(0 - 3)| / 2

Calculating step by step:

First term: 0*(3 - 0) = 0*3 = 0

Second term: 2*(0 - 0) = 2*0 = 0

Third term: 10*(0 - 3) = 10*(-3) = -30

Adding them up: 0 + 0 + (-30) = -30

Take absolute value and divide by 2: | -30 | / 2 = 30 / 2 = 15

So, the area is 15 cm². Hmm, that seems straightforward. But wait, let me verify with another method to ensure I didn't make a mistake.

Alternatively, since the triangle has vertices at (0,0), (2,3), and (10,0), maybe I can split it into simpler shapes or use base and height.

Looking at the coordinates, points A(0,0) and C(10,0) are on the x-axis, so the base AC is 10 cm. But the triangle isn't necessarily with base AC and height from B to AC. Wait, point B is at (2,3). If I consider AC as the base, which is 10 cm, then the height would be the vertical distance from point B to the base AC. However, since AC is along the x-axis, the y-coordinate of B is 3 cm. Wait, but that would make the area (base * height)/2 = (10 * 3)/2 = 15 cm². That gives the same result as the determinant method! So that's reassuring.

But hold on, is that correct? Because the triangle is not necessarily a right triangle. Wait, point B is at (2,3), so if I drop a perpendicular from B to AC (which is the x-axis), the height is indeed 3 cm, since the y-coordinate is 3. So the height is 3 cm. So area is (10*3)/2 = 15 cm². So both methods give the same result.

But wait, but in the diagram, there is a rectangle with sides 10 cm and 3 cm. The shaded triangle is inside this rectangle. If the triangle has area 15 cm², then since the rectangle is 10*3=30 cm², the triangle would be exactly half of the rectangle. But is that true? Because the triangle connects (0,0), (2,3), (10,0). Wait, but if you connect (0,0) to (2,3) to (10,0), then the triangle is not the entire half of the rectangle. Wait, but if you connect (0,0) to (10,0) to (0,3) to (0,0), that's a triangle with area (10*3)/2 = 15 cm². But the triangle in question is different. Wait, but according to our calculation, it's also 15 cm². So maybe it's equivalent? Wait, let me think.

Wait, let me visualize again. The rectangle is from (0,0) to (10,0) to (10,3) to (0,3). So the rectangle is 10 units long and 3 units high. The shaded triangle is formed by connecting (0,0), (2,3), and (10,0). So imagine the triangle that starts at the bottom left corner, goes up to a point 2 units along the top edge (since the top edge is from (0,3) to (10,3), and the point is at (2,3)), then goes down to the bottom right corner. So this triangle is not the same as the triangle formed by (0,0), (0,3), (10,0), which would be a triangle with area 15 cm². But according to our calculation, the triangle (0,0), (2,3), (10,0) also has area 15 cm². That seems counterintuitive. How can two different triangles have the same area? Well, maybe they do. Let me check with another approach.

Alternatively, maybe we can use coordinates to calculate the area. Let's consider coordinates again. If I split the triangle into parts or use coordinates. Another way to calculate the area is using the shoelace formula, which is the same as the determinant formula.

Shoelace formula:

List the coordinates in order, repeating the first at the end:

(0,0), (2,3), (10,0), (0,0)

Compute sum of products downward:

(0*3) + (2*0) + (10*0) = 0 + 0 + 0 = 0

Compute sum of products upward:

(0*2) + (3*10) + (0*0) = 0 + 30 + 0 = 30

Area = |0 - 30| / 2 = 30 / 2 = 15 cm². Same result.

Hmm. So according to this, the area is 15 cm². But let me think again. Maybe the confusion comes from the rectangle. If we think about the triangle from (0,0), (10,0), (0,3), that's 15 cm², but the triangle in the problem is a different triangle. However, the area is the same. Is that possible?

Alternatively, maybe there is a mistake in my reasoning. Let me think. If I imagine the line from (0,0) to (2,3), then from (2,3) to (10,0). So this triangle is cutting off a part of the rectangle. Let me try to compute the area by subtracting some areas from the rectangle.

Alternatively, maybe using coordinates again. Let me try to find the equations of the lines and see.

First, the line from (0,0) to (2,3). The slope is (3 - 0)/(2 - 0) = 3/2. So the equation is y = (3/2)x.

Then, the line from (2,3) to (10,0). The slope is (0 - 3)/(10 - 2) = (-3)/8. So the equation is y - 3 = (-3/8)(x - 2). So y = (-3/8)x + (3/8)*2 + 3 = (-3/8)x + 6/8 + 24/8 = (-3/8)x + 30/8 = (-3/8)x + 15/4.

Now, if I want to find the area of the triangle, I can integrate or something, but that seems overkill. Alternatively, maybe find the intersection points or something else.

Alternatively, since we already have two methods giving 15 cm², maybe that's the answer. Wait, but why does the triangle have the same area as half the rectangle? Maybe the line from (0,0) to (2,3) divides the rectangle in some way?

Wait, but the rectangle has area 30 cm². If the triangle has area 15 cm², it's exactly half of the rectangle. Is there a reason for that? Maybe. For example, if you connect (0,0) to (2,3) to (10,0), maybe this triangle is congruent to another triangle in the rectangle? Or maybe it's symmetrical?

Alternatively, maybe not. Wait, let me think of coordinates. Let me try to use coordinate geometry to find the area between the lines. Wait, perhaps decomposing the figure.

Alternatively, if I consider the triangle ABC with A(0,0), B(2,3), C(10,0). The base AC is 10 cm, as before. The height is not 3 cm, because the height is the perpendicular distance from B to AC. Wait, earlier I thought that since AC is on the x-axis, the y-coordinate of B is 3, so that's the height. But is that correct?

Wait, if AC is along the x-axis from (0,0) to (10,0), then the line AC is y = 0. The height from point B(2,3) to AC is indeed the vertical distance from B to the line AC, which is 3 units. Therefore, the height is 3 cm. Therefore, area is (10*3)/2 = 15 cm². That seems correct.

But then why does the triangle (0,0), (2,3), (10,0) have the same area as (0,0), (0,3), (10,0)? Maybe because they are both triangles with the same base and same height? Wait, no. Wait, triangle (0,0), (0,3), (10,0) has base AC = 10 cm and height 3 cm. Triangle (0,0), (2,3), (10,0) also has base AC = 10 cm and height 3 cm? How is that possible? Wait, if the height is measured perpendicularly from the base AC, then regardless of where the third point is, as long as its vertical distance from AC is 3 cm, then the area would be same. But in this case, point (2,3) is vertically 3 cm above AC, yes. So even though it's not directly above the midpoint, the vertical distance is still 3 cm. Therefore, area is (base * height)/2. Hence 15 cm².

Therefore, maybe that is correct. However, let me try to think of another way to confirm.

Alternatively, perhaps using vectors. The vectors from A to B and from A to C.

Vector AB = (2,3) - (0,0) = (2,3)

Vector AC = (10,0) - (0,0) = (10,0)

The area of the triangle is half the magnitude of the cross product of AB and AC.

Cross product in 2D is scalar magnitude |Ax*By - Ay*Bx|. Wait, more accurately, for vectors AB and AC, the cross product is AB_x * AC_y - AB_y * AC_x.

So AB is (2,3), AC is (10,0). So cross product is 2*0 - 3*10 = 0 - 30 = -30. The magnitude is 30, so area is 30/2 = 15 cm². Yep, same answer again.

So all methods point to 15 cm². Therefore, maybe 15 is the correct answer.

But wait, the Asymptote code labels the rectangle with "10 cm" at the top and bottom, and "3 cm" on the left and right sides. So the rectangle is 10 cm by 3 cm. The triangle is inside this rectangle. The question is, is the area 15 cm²? That would be half of the rectangle. But let me think again. Suppose I have a rectangle of area 30 cm², and the triangle is half of that. Is that possible?

Yes, for example, if you draw a diagonal of the rectangle, you get two triangles each with area 15 cm². However, in this case, the triangle isn't a diagonal. It's connecting (0,0) to (2,3) to (10,0). But since the vertical distance from (2,3) to the base is 3 cm, same as the height of the rectangle, then the area would be same as the triangle formed by the diagonal. Wait, but the diagonal would be from (0,0) to (10,3), which is a different point. So why is the area same?

Wait, maybe because the average of the heights? Wait, not sure. But according to the calculations, yes.

Alternatively, maybe there's a misunderstanding. Let me check coordinates again.

If I imagine the rectangle, from (0,0) to (10,0) to (10,3) to (0,3). The shaded triangle is connecting (0,0) to (2,3) to (10,0). So this triangle is actually occupying the same area as the triangle connecting (0,0) to (10,0) to (0,3). Because both have base 10 cm and height 3 cm.

But how? Because the third point in the first triangle is (2,3), which is on the top edge of the rectangle. So maybe the area is same because regardless of where you put the third point on the top edge, if you connect it to (0,0) and (10,0), the area remains same?

Wait, let me test with another point. Suppose the third point is (5,3). Then the triangle would be (0,0), (5,3), (10,0). What is its area? Using the determinant formula:

Ax=0, Ay=0

Bx=5, By=3

Cx=10, Cy=0

Area = |0*(3 - 0) + 5*(0 - 0) + 10*(0 - 3)| / 2 = |0 + 0 + (-30)| / 2 = 30/2 = 15 cm². Same result.

So if you pick any point on the top edge of the rectangle (from (0,3) to (10,3)), connecting it to (0,0) and (10,0) gives a triangle of area 15 cm². That's interesting. So why is that?

Because the area of such a triangle is (base * height)/2, where base is 10 cm, and the height is 3 cm. Since the height is the vertical distance from the top edge to the base, which is 3 cm. So regardless of where the point is on the top edge, the height remains 3 cm. Therefore, the area remains same.

Therefore, even though the triangle is not a diagonal, it's similar in area.

So, in our problem, the third point is at (2,3), so it's 2 units from the left on the top edge, but the area is still 15 cm². Therefore, the answer is 15 cm².

But let me check with coordinates once more, just to make sure. Maybe drawing a diagram mentally. The triangle has vertices at (0,0), (2,3), and (10,0). If I imagine this triangle, it will have a base from (0,0) to (10,0), which is 10 cm. The third vertex is somewhere above the base, at (2,3). The height here is 3 cm as before. So, yes, the area is 1/2 * 10 * 3 = 15 cm².

Alternatively, I can think of the triangle as part of the rectangle. If I imagine the rectangle divided by the line from (0,0) to (2,3) and then from (2,3) to (10,0). Maybe we can find some other way to calculate the area. For example, breaking the rectangle into different regions.

Let me try that. The rectangle is 10 cm by 3 cm. Let's think of the triangle as being composed of two regions or something.

First, from (0,0) to (2,3) to (10,0). Let me see. Alternatively, perhaps compute the area of the triangle by integrating or using coordinates.

Alternatively, since I have two different methods giving me the same answer, and my reasoning checks out, maybe 15 cm² is correct.

But wait, let me think again. Let me check using base and height. Suppose I take another base. For instance, take the base as the segment from (0,0) to (2,3). Then I need to find the corresponding height.

But that might be more complicated. Let me try. The length of AB is sqrt((2-0)^2 + (3-0)^2) = sqrt(4 + 9) = sqrt(13). Then, to find the height corresponding to this base, I need the height from point C(10,0) to the line AB.

First, find the equation of line AB. As earlier, it's y = (3/2)x.

The distance from point C(10,0) to this line is |(3/2)(10) - 0 + 0| / sqrt((3/2)^2 + (-1)^2) ?

Wait, general formula for distance from point (x0, y0) to line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a² + b²). So let me rewrite the equation of AB in ax + by + c = 0 form.

Original line AB: y = (3/2)x => (3/2)x - y = 0. Multiply both sides by 2 to eliminate fraction: 3x - 2y = 0. So a = 3, b = -2, c = 0.

Distance from (10, 0):

|3*10 + (-2)*0 + 0| / sqrt(3² + (-2)^2) = |30| / sqrt(9 + 4) = 30 / sqrt(13).

Therefore, area of triangle is (base * height)/2 = (sqrt(13) * 30 / sqrt(13))/2 = 30/2 = 15 cm².

Same result again. Therefore, regardless of which base I choose, the area is 15 cm².

Therefore, I think I can be confident that the area is 15 cm².

But just to make sure, let me think of another approach. Maybe coordinate geometry with vectors.

The area can also be calculated by using coordinates and determinants. As done earlier, which gives 15.

Alternatively, maybe using matrices.

But I think all approaches confirm that the area is 15 cm².

Therefore, I think the answer is 15 cm².

**Final Answer**
The area of the shaded triangle is \boxed{15} square centimeters.

To determine the area of the shaded triangle shown in the diagram, we are provided with coordinates of its vertices:

- $ A = (0, 0) $
- $ B = (2, 3) $
- $ C = (10, 0) $

We are given that the rectangle has a base of 10 cm and height of 3 cm, and the shaded triangle lies entirely within this rectangle. The triangle is formed by connecting the points $ A $, $ B $, and $ C $.

---

### Step 1: Use the Coordinate Determinant Formula

The formula for the area of a triangle given its vertices $ (x_1, y_1) $, $ (x_2, y_2) $, and $ (x_3, y_3) $ is:

$$
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
$$

Substituting the coordinates of the triangle:

$$
\text{Area} = \frac{1}{2} \left| 0(3 - 0) + 2(0 - 0) + 10(0 - 3) \right|
$$
$$
= \frac{1}{2} \left| 0 + 0 - 30 \right| = \frac{1}{2} \times 30 = 15
$$

So the area of the triangle is **15 cm²**.

---

### Step 2: Reconfirm Using Base and Height

Let's consider the base of the triangle as the segment from $ A = (0, 0) $ to $ C = (10, 0) $. This forms a horizontal base of length 10 cm.

The height corresponds to the vertical distance from the third vertex $ B = (2, 3) $ to this base. Since the base lies along the x-axis, the y-coordinate of point $ B $ gives the height, which is:

$$
\text{Height} = 3 \text{ cm}
$$

Now, apply the area formula using base and height:

$$
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 3 = 15 \text{ cm}^2
$$

---

### Step 3: Geometric Insight (Optional)

The diagram shows that the triangle lies within a rectangle of area $ 10 \times 3 = 30 $ cm². If the triangle uses the same vertical height (3 cm) and base (10 cm), and its third vertex lies on the top edge of the rectangle, then the triangle's area is **exactly half the area of the rectangle**—regardless of where the third vertex is placed on the top edge, as long as the height remains 3 cm.

This provides an intuitive way to understand why the triangle’s area is consistent at 15 cm².

---

### Final Result

All approaches confirm the same result.

$$
\boxed{15}
$$

Solution: The shaded triangle has a base of length $10\text{ cm}.$ Since the triangle is enclosed in a rectangle of height $3\text{ cm},$ then the height of the triangle is $3\text{ cm}.$ (We know that the enclosing shape is a rectangle, because any figure with four sides, including two pairs of equal opposite sides, and two right angles must be a rectangle.) Therefore, the area of the triangle is $$\frac{1}{2}\times 3 \times 10 = \boxed{15\mbox{ cm}^2}.$$