Explanation:
Okay, so I need to figure out how many sides a convex polygon has if the sum of all but one of its interior angles is 1070 degrees. Hmm, let me start by recalling some formulas related to polygons.

First, I remember that the sum of the interior angles of a convex polygon with n sides is given by the formula (n - 2) * 180 degrees. That seems right. For example, a triangle (3 sides) has (3 - 2)*180 = 180 degrees, which matches. A quadrilateral (4 sides) has (4 - 2)*180 = 360 degrees, which also checks out. So that formula is solid.

Now, the problem says that the sum of all but one of the interior angles is 1070 degrees. So, if the total sum of all interior angles is S, then the sum of all but one would be S minus one of the angles. Let me denote the measure of the missing angle as x. Therefore, we have:

S - x = 1070 degrees.

But S is also equal to (n - 2)*180. So substituting that in:

(n - 2)*180 - x = 1070.

But I don't know the value of x. However, since the polygon is convex, all interior angles must be less than 180 degrees. That might be useful later.

Wait, but maybe I can express x in terms of n? Let me think. If I rearrange the equation:

x = (n - 2)*180 - 1070.

But since x is an interior angle of a convex polygon, it must satisfy 0 < x < 180 degrees. Therefore, we can set up inequalities:

0 < (n - 2)*180 - 1070 < 180.

So, solving these inequalities might help me find the value of n. Let me write that down:

First inequality:

(n - 2)*180 - 1070 > 0

=> (n - 2)*180 > 1070

=> n - 2 > 1070 / 180

Let me compute 1070 divided by 180. Let me do that division.

Well, 180*5 = 900, 180*6 = 1080. So 1070 is 10 less than 1080. Therefore, 1070 / 180 = 6 - 10/180 = 6 - 1/18 ≈ 5.944...

So, n - 2 > approximately 5.944, so n > 5.944 + 2 = 7.944. Since n must be an integer (number of sides), n must be at least 8.

Now the second inequality:

(n - 2)*180 - 1070 < 180

=> (n - 2)*180 < 1070 + 180 = 1250

=> n - 2 < 1250 / 180

Calculating 1250 / 180. Let me divide numerator and denominator by 10: 125 / 18 ≈ 6.944...

So, n - 2 < 6.944, so n < 6.944 + 2 = 8.944. Since n must be an integer, n must be less than 8.944, so n ≤ 8.

Wait, so combining both inequalities: n must be greater than approximately 7.944 and less than approximately 8.944. Since n is an integer, the only possible value is n = 8.

So, the polygon has 8 sides. Let me check if this makes sense.

Let me compute the total sum of interior angles for n = 8: (8 - 2)*180 = 6*180 = 1080 degrees. Then, the sum of all but one angle is 1070 degrees, so the missing angle is 1080 - 1070 = 10 degrees. Wait, 10 degrees? But in a convex polygon, each interior angle must be less than 180 degrees. 10 degrees is way less than 180, so that's okay. But is 10 degrees a valid interior angle? Well, technically, yes. Although it's a very "sharp" angle, but since the polygon is convex, all angles just need to be less than 180. So, 10 degrees is acceptable.

But let me check if there's another possible value. Wait, according to the inequalities, n must be 8. But let me check n=7. If n=7, total sum is (7-2)*180=5*180=900 degrees. Then, the sum of all but one angle would be 900 - x = 1070? Wait, that would mean x = 900 - 1070 = -70 degrees. Which is impossible, since angles can't be negative. So n=7 is invalid. Similarly, n=9: total sum is (9-2)*180=7*180=1260. Then, the missing angle would be 1260 - 1070 = 90 degrees. Which is also valid, since 90 < 180. Wait, but according to our previous calculation, n must be less than 8.944, so n=9 would be 9, which is more than 8.944? Wait, 8.944 is approximately 8.944, so n must be less than that. So n=9 would be 9, which is more than 8.944, so it's not allowed. But according to the second inequality, n - 2 < 1250 / 180 ≈ 6.944, so n < 8.944. Therefore, n=9 is not allowed. But if n=9, then the missing angle is 90 degrees, which is okay. But according to the inequality, n=9 is not allowed. So why is that?

Wait, maybe my inequalities were wrong? Let me check again.

Original equation:

x = (n - 2)*180 - 1070

And since x must be between 0 and 180, exclusive.

So:

0 < (n - 2)*180 - 1070 < 180

So first inequality:

(n - 2)*180 - 1070 > 0 => (n - 2)*180 > 1070 => n - 2 > 1070 / 180 ≈ 5.944 => n > 7.944 => n ≥ 8

Second inequality:

(n - 2)*180 - 1070 < 180 => (n - 2)*180 < 1250 => n - 2 < 1250 / 180 ≈ 6.944 => n < 8.944 => n ≤ 8

Therefore, n must be 8. So n=9 is not allowed. But if n=9, then x = 1260 - 1070 = 90, which is valid. But according to the inequality, n=9 is not allowed. So why is that?

Wait, maybe my mistake is in the second inequality? Let me check again.

The second inequality is:

(n - 2)*180 - 1070 < 180

Which is equivalent to:

(n - 2)*180 < 1070 + 180 = 1250

So, (n - 2) < 1250 / 180 ≈ 6.944...

So n - 2 < 6.944, so n < 8.944. Therefore, n can be 8, since 8 < 8.944. But n=9 would be 9, which is greater than 8.944, so it's not allowed. However, if n=9, then x=90, which is less than 180. So why is the inequality not allowing n=9?

Wait, maybe my mistake is that I considered x < 180, but maybe there's another constraint? Let me think. The problem says the polygon is convex, so all interior angles must be less than 180. Therefore, x must be less than 180. However, if n=9, then x=90, which is less than 180, so that's okay. But according to the inequality, n=9 is not allowed. Therefore, there's a contradiction here. Which means my reasoning is wrong somewhere.

Wait, let me check the second inequality again. If n=9, then:

(n - 2)*180 - 1070 = 7*180 - 1070 = 1260 - 1070 = 90, which is less than 180. So, according to the inequality, (n - 2)*180 - 1070 < 180 is true for n=9. But according to my calculation, n < 8.944. Wait, 1250 / 180 is approximately 6.944, so n - 2 < 6.944 => n < 8.944. But n=9 is 9, which is greater than 8.944. But 1250 / 180 is exactly 6.944... So, if n - 2 is less than 6.944..., then n is less than 8.944... So n=9 is not less than 8.944... So why does n=9 satisfy the inequality?

Wait, let me compute 1250 / 180 exactly. Let me divide 1250 by 180:

Divide numerator and denominator by 10: 125 / 18. Let me compute 18*6 = 108, 18*6.944... = 125. So 125 / 18 is exactly equal to 6 + 17/18 ≈ 6.944... So, n - 2 < 6 + 17/18. Therefore, n < 8 + 17/18 ≈ 8.944... So n must be less than that. Therefore, n=8 is allowed, but n=9 is not. However, when n=9, (n - 2)*180 - 1070 = 90, which is less than 180. So why is that?

Wait, maybe my mistake is that I set up the inequality as (n - 2)*180 - 1070 < 180, but actually, the missing angle x is equal to (n - 2)*180 - 1070, and since x must be less than 180, then:

(n - 2)*180 - 1070 < 180

But if n=9, then (n - 2)*180 - 1070 = 90 < 180, which is true. However, according to the inequality, n - 2 < 1250 / 180 ≈ 6.944, so n < 8.944. But n=9 is not less than 8.944. Therefore, there's a contradiction here. Wait, but if n=9, then n - 2 = 7, and 7*180 = 1260. Then 1260 - 1070 = 90 < 180. So why does the inequality say n - 2 < 6.944? Because 1250 / 180 is approximately 6.944, but n - 2 is 7, which is greater than 6.944. Therefore, the inequality (n - 2)*180 < 1250 is not satisfied for n=9. Wait, but (n - 2)*180 = 7*180 = 1260, which is greater than 1250. Therefore, the inequality (n - 2)*180 < 1250 is not satisfied. Therefore, for n=9, the value of (n - 2)*180 - 1070 = 90, which is less than 180, but the inequality (n - 2)*180 < 1250 is not satisfied. Therefore, there's a mistake here.

Wait, so my mistake was in the second inequality. Let me re-examine that.

Original equation:

x = (n - 2)*180 - 1070

And x must be less than 180. Therefore:

(n - 2)*180 - 1070 < 180

Which rearranges to:

(n - 2)*180 < 1070 + 180 = 1250

Therefore, (n - 2) < 1250 / 180 ≈ 6.944...

Therefore, n - 2 must be less than approximately 6.944, so n < 8.944. Therefore, n=8 is the only integer value. However, when n=9, (n - 2)*180 = 1260, which is greater than 1250, so the inequality is not satisfied. Therefore, even though x=90 is less than 180, the value of (n - 2)*180 is greater than 1250, which would mean that x = 1260 - 1070 = 90, which is less than 180. But according to the inequality, (n - 2)*180 must be less than 1250. Therefore, there's a contradiction here.

Wait, so why is that? Because the inequality is derived from x < 180. So if x = (n - 2)*180 - 1070 < 180, then (n - 2)*180 < 1250. Therefore, if (n - 2)*180 is greater than 1250, then x would be greater than 180? Wait, no. Wait, if (n - 2)*180 is greater than 1250, then (n - 2)*180 - 1070 would be greater than 1250 - 1070 = 180. Therefore, x would be greater than 180. But in the case of n=9, (n - 2)*180 = 1260, so x = 1260 - 1070 = 90, which is less than 180. Wait, that contradicts.

Wait, hold on. Let me do the math again. If (n - 2)*180 < 1250, then x = (n - 2)*180 - 1070 < 1250 - 1070 = 180. Therefore, if (n - 2)*180 is less than 1250, then x is less than 180. However, if (n - 2)*180 is greater than 1250, then x = (n - 2)*180 - 1070 would be greater than 1250 - 1070 = 180. Therefore, x would be greater than 180. Therefore, if we want x < 180, we need (n - 2)*180 < 1250. Therefore, if (n - 2)*180 is greater than 1250, then x would be greater than 180, which is not allowed in a convex polygon. Therefore, for n=9, (n - 2)*180 = 1260, which is greater than 1250, so x = 1260 - 1070 = 90, which is less than 180. Wait, that contradicts the previous conclusion.

Wait, there's a mistake here. Let me check again. If (n - 2)*180 is greater than 1250, then (n - 2)*180 - 1070 is greater than 1250 - 1070 = 180? Wait, no. If A > B, then A - C > B - C. So if (n - 2)*180 > 1250, then (n - 2)*180 - 1070 > 1250 - 1070 = 180. Therefore, x > 180. But in the case of n=9, (n - 2)*180 = 1260, so x = 1260 - 1070 = 90, which is less than 180. Wait, that's not matching.

Wait, 1260 - 1070 is 190? Wait, no! Wait, 1260 - 1070. Let me compute that again. 1260 - 1000 = 260, then subtract 70 more: 260 - 70 = 190. Wait, so x = 190 degrees? But earlier I thought it was 90. Wait, that's my mistake! I think I miscalculated earlier.

Wait, if n=9, total sum is (9 - 2)*180 = 7*180 = 1260. Then, the sum of all but one angle is 1070, so the missing angle is 1260 - 1070 = 190 degrees. Which is greater than 180 degrees. But in a convex polygon, all interior angles must be less than 180 degrees. Therefore, 190 degrees is invalid. Therefore, n=9 is invalid.

Oh! I made a mistake earlier when I thought that 1260 - 1070 was 90. Actually, it's 190. So that's why the inequality is important. Therefore, n=9 would result in an interior angle of 190 degrees, which is not allowed in a convex polygon. Therefore, n=9 is invalid. Therefore, only n=8 is valid.

So, going back, when n=8, total sum is 6*180=1080. Then, the missing angle is 1080 - 1070 = 10 degrees, which is valid. Therefore, n=8 is the correct answer.

So, my initial calculation was correct. The confusion was due to a miscalculation when I thought 1260 - 1070 was 90, but it's actually 190. Therefore, n=9 is invalid because it would require an interior angle of 190 degrees, which is not allowed in a convex polygon. Therefore, the only possible value is n=8.

Let me just check n=8 again. Total sum is 1080. Sum of all but one is 1070, so missing angle is 10 degrees. Since 10 < 180, it's valid. Therefore, n=8 is the answer.

But just to make sure there are no other possible values. For example, n=7: total sum is 5*180=900. Then, sum of all but one is 900 - x = 1070 => x = 900 - 1070 = -170, which is invalid. n=8 is valid. n=9 is invalid. n=10: total sum is 8*180=1440. Then, missing angle would be 1440 - 1070 = 370 degrees, which is way more than 180, invalid. So, no. Therefore, only n=8 is valid.

Therefore, the answer is 8 sides.

**Final Answer**
The convex polygon has \boxed{8} sides.

To determine the number of sides in a convex polygon where the sum of all but one of its interior angles is $1070^\circ$, we begin by recalling the formula for the sum of the interior angles of a convex polygon with $n$ sides:

$$
\text{Sum of interior angles} = (n - 2) \times 180^\circ
$$

Let the missing interior angle be $x$. Then, the sum of all but one of the interior angles is:

$$
(n - 2) \times 180 - x = 1070
$$

From this, we can solve for $x$:

$$
x = (n - 2) \times 180 - 1070
$$

Since the polygon is convex, all interior angles must be **less than $180^\circ$**. Therefore, we require:

$$
0 < x < 180
$$

Substituting the expression for $x$:

$$
0 < (n - 2) \times 180 - 1070 < 180
$$

---

### Step 1: Solve the Inequalities

**First inequality:**

$$
(n - 2) \times 180 - 1070 > 0
$$
$$
(n - 2) \times 180 > 1070
$$
$$
n - 2 > \frac{1070}{180} \approx 5.944
$$
$$
n > 7.944
$$

**Second inequality:**

$$
(n - 2) \times 180 - 1070 < 180
$$
$$
(n - 2) \times 180 < 1250
$$
$$
n - 2 < \frac{1250}{180} \approx 6.944
$$
$$
n < 8.944
$$

Combining both inequalities:

$$
7.944 < n < 8.944
$$

Since $n$ must be an integer, the only possible value is:

$$
n = 8
$$

---

### Step 2: Verify the Solution

For $n = 8$, the total sum of interior angles is:

$$
(8 - 2) \times 180 = 6 \times 180 = 1080^\circ
$$

If the sum of all but one angle is $1070^\circ$, then the missing angle is:

$$
x = 1080 - 1070 = 10^\circ
$$

Since $10^\circ < 180^\circ$, this is a valid interior angle for a convex polygon.

---

### Step 3: Check for Other Possible Values

- For $n = 7$: Total sum = $5 \times 180 = 900^\circ$, so missing angle = $900 - 1070 = -170^\circ$ (invalid).
- For $n = 9$: Total sum = $7 \times 180 = 1260^\circ$, so missing angle = $1260 - 1070 = 190^\circ$ (invalid, as it exceeds $180^\circ$).

Thus, only $n = 8$ satisfies all conditions.

---

### Final Answer

$$
\boxed{8}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many sides a convex polygon has if the sum of all but one of its interior angles is 1070 degrees. Hmm, let me start by recalling some formulas related to polygons.

First, I remember that the sum of the interior angles of a convex polygon with n sides is given by the formula (n - 2) * 180 degrees. That seems right. For example, a triangle (3 sides) has (3 - 2)*180 = 180 degrees, which matches. A quadrilateral (4 sides) has (4 - 2)*180 = 360 degrees, which also checks out. So that formula is solid.

Now, the problem says that the sum of all but one of the interior angles is 1070 degrees. So, if the total sum of all interior angles is S, then the sum of all but one would be S minus one of the angles. Let me denote the measure of the missing angle as x. Therefore, we have:

S - x = 1070 degrees.

But S is also equal to (n - 2)*180. So substituting that in:

(n - 2)*180 - x = 1070.

But I don't know the value of x. However, since the polygon is convex, all interior angles must be less than 180 degrees. That might be useful later.

Wait, but maybe I can express x in terms of n? Let me think. If I rearrange the equation:

x = (n - 2)*180 - 1070.

But since x is an interior angle of a convex polygon, it must satisfy 0 < x < 180 degrees. Therefore, we can set up inequalities:

0 < (n - 2)*180 - 1070 < 180.

So, solving these inequalities might help me find the value of n. Let me write that down:

First inequality:

(n - 2)*180 - 1070 > 0

=> (n - 2)*180 > 1070

=> n - 2 > 1070 / 180

Let me compute 1070 divided by 180. Let me do that division.

Well, 180*5 = 900, 180*6 = 1080. So 1070 is 10 less than 1080. Therefore, 1070 / 180 = 6 - 10/180 = 6 - 1/18 ≈ 5.944...

So, n - 2 > approximately 5.944, so n > 5.944 + 2 = 7.944. Since n must be an integer (number of sides), n must be at least 8.

Now the second inequality:

(n - 2)*180 - 1070 < 180

=> (n - 2)*180 < 1070 + 180 = 1250

=> n - 2 < 1250 / 180

Calculating 1250 / 180. Let me divide numerator and denominator by 10: 125 / 18 ≈ 6.944...

So, n - 2 < 6.944, so n < 6.944 + 2 = 8.944. Since n must be an integer, n must be less than 8.944, so n ≤ 8.

Wait, so combining both inequalities: n must be greater than approximately 7.944 and less than approximately 8.944. Since n is an integer, the only possible value is n = 8.

So, the polygon has 8 sides. Let me check if this makes sense.

Let me compute the total sum of interior angles for n = 8: (8 - 2)*180 = 6*180 = 1080 degrees. Then, the sum of all but one angle is 1070 degrees, so the missing angle is 1080 - 1070 = 10 degrees. Wait, 10 degrees? But in a convex polygon, each interior angle must be less than 180 degrees. 10 degrees is way less than 180, so that's okay. But is 10 degrees a valid interior angle? Well, technically, yes. Although it's a very "sharp" angle, but since the polygon is convex, all angles just need to be less than 180. So, 10 degrees is acceptable.

But let me check if there's another possible value. Wait, according to the inequalities, n must be 8. But let me check n=7. If n=7, total sum is (7-2)*180=5*180=900 degrees. Then, the sum of all but one angle would be 900 - x = 1070? Wait, that would mean x = 900 - 1070 = -70 degrees. Which is impossible, since angles can't be negative. So n=7 is invalid. Similarly, n=9: total sum is (9-2)*180=7*180=1260. Then, the missing angle would be 1260 - 1070 = 90 degrees. Which is also valid, since 90 < 180. Wait, but according to our previous calculation, n must be less than 8.944, so n=9 would be 9, which is more than 8.944? Wait, 8.944 is approximately 8.944, so n must be less than that. So n=9 would be 9, which is more than 8.944, so it's not allowed. But according to the second inequality, n - 2 < 1250 / 180 ≈ 6.944, so n < 8.944. Therefore, n=9 is not allowed. But if n=9, then the missing angle is 90 degrees, which is okay. But according to the inequality, n=9 is not allowed. So why is that?

Wait, maybe my inequalities were wrong? Let me check again.

Original equation:

x = (n - 2)*180 - 1070

And since x must be between 0 and 180, exclusive.

So:

0 < (n - 2)*180 - 1070 < 180

So first inequality:

(n - 2)*180 - 1070 > 0 => (n - 2)*180 > 1070 => n - 2 > 1070 / 180 ≈ 5.944 => n > 7.944 => n ≥ 8

Second inequality:

(n - 2)*180 - 1070 < 180 => (n - 2)*180 < 1250 => n - 2 < 1250 / 180 ≈ 6.944 => n < 8.944 => n ≤ 8

Therefore, n must be 8. So n=9 is not allowed. But if n=9, then x = 1260 - 1070 = 90, which is valid. But according to the inequality, n=9 is not allowed. So why is that?

Wait, maybe my mistake is in the second inequality? Let me check again.

The second inequality is:

(n - 2)*180 - 1070 < 180

Which is equivalent to:

(n - 2)*180 < 1070 + 180 = 1250

So, (n - 2) < 1250 / 180 ≈ 6.944...

So n - 2 < 6.944, so n < 8.944. Therefore, n can be 8, since 8 < 8.944. But n=9 would be 9, which is greater than 8.944, so it's not allowed. However, if n=9, then x=90, which is less than 180. So why is the inequality not allowing n=9?

Wait, maybe my mistake is that I considered x < 180, but maybe there's another constraint? Let me think. The problem says the polygon is convex, so all interior angles must be less than 180. Therefore, x must be less than 180. However, if n=9, then x=90, which is less than 180, so that's okay. But according to the inequality, n=9 is not allowed. Therefore, there's a contradiction here. Which means my reasoning is wrong somewhere.

Wait, let me check the second inequality again. If n=9, then:

(n - 2)*180 - 1070 = 7*180 - 1070 = 1260 - 1070 = 90, which is less than 180. So, according to the inequality, (n - 2)*180 - 1070 < 180 is true for n=9. But according to my calculation, n < 8.944. Wait, 1250 / 180 is approximately 6.944, so n - 2 < 6.944 => n < 8.944. But n=9 is 9, which is greater than 8.944. But 1250 / 180 is exactly 6.944... So, if n - 2 is less than 6.944..., then n is less than 8.944... So n=9 is not less than 8.944... So why does n=9 satisfy the inequality?

Wait, let me compute 1250 / 180 exactly. Let me divide 1250 by 180:

Divide numerator and denominator by 10: 125 / 18. Let me compute 18*6 = 108, 18*6.944... = 125. So 125 / 18 is exactly equal to 6 + 17/18 ≈ 6.944... So, n - 2 < 6 + 17/18. Therefore, n < 8 + 17/18 ≈ 8.944... So n must be less than that. Therefore, n=8 is allowed, but n=9 is not. However, when n=9, (n - 2)*180 - 1070 = 90, which is less than 180. So why is that?

Wait, maybe my mistake is that I set up the inequality as (n - 2)*180 - 1070 < 180, but actually, the missing angle x is equal to (n - 2)*180 - 1070, and since x must be less than 180, then:

(n - 2)*180 - 1070 < 180

But if n=9, then (n - 2)*180 - 1070 = 90 < 180, which is true. However, according to the inequality, n - 2 < 1250 / 180 ≈ 6.944, so n < 8.944. But n=9 is not less than 8.944. Therefore, there's a contradiction here. Wait, but if n=9, then n - 2 = 7, and 7*180 = 1260. Then 1260 - 1070 = 90 < 180. So why does the inequality say n - 2 < 6.944? Because 1250 / 180 is approximately 6.944, but n - 2 is 7, which is greater than 6.944. Therefore, the inequality (n - 2)*180 < 1250 is not satisfied for n=9. Wait, but (n - 2)*180 = 7*180 = 1260, which is greater than 1250. Therefore, the inequality (n - 2)*180 < 1250 is not satisfied. Therefore, for n=9, the value of (n - 2)*180 - 1070 = 90, which is less than 180, but the inequality (n - 2)*180 < 1250 is not satisfied. Therefore, there's a mistake here.

Wait, so my mistake was in the second inequality. Let me re-examine that.

Original equation:

x = (n - 2)*180 - 1070

And x must be less than 180. Therefore:

(n - 2)*180 - 1070 < 180

Which rearranges to:

(n - 2)*180 < 1070 + 180 = 1250

Therefore, (n - 2) < 1250 / 180 ≈ 6.944...

Therefore, n - 2 must be less than approximately 6.944, so n < 8.944. Therefore, n=8 is the only integer value. However, when n=9, (n - 2)*180 = 1260, which is greater than 1250, so the inequality is not satisfied. Therefore, even though x=90 is less than 180, the value of (n - 2)*180 is greater than 1250, which would mean that x = 1260 - 1070 = 90, which is less than 180. But according to the inequality, (n - 2)*180 must be less than 1250. Therefore, there's a contradiction here.

Wait, so why is that? Because the inequality is derived from x < 180. So if x = (n - 2)*180 - 1070 < 180, then (n - 2)*180 < 1250. Therefore, if (n - 2)*180 is greater than 1250, then x would be greater than 180? Wait, no. Wait, if (n - 2)*180 is greater than 1250, then (n - 2)*180 - 1070 would be greater than 1250 - 1070 = 180. Therefore, x would be greater than 180. But in the case of n=9, (n - 2)*180 = 1260, so x = 1260 - 1070 = 90, which is less than 180. Wait, that contradicts.

Wait, hold on. Let me do the math again. If (n - 2)*180 < 1250, then x = (n - 2)*180 - 1070 < 1250 - 1070 = 180. Therefore, if (n - 2)*180 is less than 1250, then x is less than 180. However, if (n - 2)*180 is greater than 1250, then x = (n - 2)*180 - 1070 would be greater than 1250 - 1070 = 180. Therefore, x would be greater than 180. Therefore, if we want x < 180, we need (n - 2)*180 < 1250. Therefore, if (n - 2)*180 is greater than 1250, then x would be greater than 180, which is not allowed in a convex polygon. Therefore, for n=9, (n - 2)*180 = 1260, which is greater than 1250, so x = 1260 - 1070 = 90, which is less than 180. Wait, that contradicts the previous conclusion.

Wait, there's a mistake here. Let me check again. If (n - 2)*180 is greater than 1250, then (n - 2)*180 - 1070 is greater than 1250 - 1070 = 180? Wait, no. If A > B, then A - C > B - C. So if (n - 2)*180 > 1250, then (n - 2)*180 - 1070 > 1250 - 1070 = 180. Therefore, x > 180. But in the case of n=9, (n - 2)*180 = 1260, so x = 1260 - 1070 = 90, which is less than 180. Wait, that's not matching.

Wait, 1260 - 1070 is 190? Wait, no! Wait, 1260 - 1070. Let me compute that again. 1260 - 1000 = 260, then subtract 70 more: 260 - 70 = 190. Wait, so x = 190 degrees? But earlier I thought it was 90. Wait, that's my mistake! I think I miscalculated earlier.

Wait, if n=9, total sum is (9 - 2)*180 = 7*180 = 1260. Then, the sum of all but one angle is 1070, so the missing angle is 1260 - 1070 = 190 degrees. Which is greater than 180 degrees. But in a convex polygon, all interior angles must be less than 180 degrees. Therefore, 190 degrees is invalid. Therefore, n=9 is invalid.

Oh! I made a mistake earlier when I thought that 1260 - 1070 was 90. Actually, it's 190. So that's why the inequality is important. Therefore, n=9 would result in an interior angle of 190 degrees, which is not allowed in a convex polygon. Therefore, n=9 is invalid. Therefore, only n=8 is valid.

So, going back, when n=8, total sum is 6*180=1080. Then, the missing angle is 1080 - 1070 = 10 degrees, which is valid. Therefore, n=8 is the correct answer.

So, my initial calculation was correct. The confusion was due to a miscalculation when I thought 1260 - 1070 was 90, but it's actually 190. Therefore, n=9 is invalid because it would require an interior angle of 190 degrees, which is not allowed in a convex polygon. Therefore, the only possible value is n=8.

Let me just check n=8 again. Total sum is 1080. Sum of all but one is 1070, so missing angle is 10 degrees. Since 10 < 180, it's valid. Therefore, n=8 is the answer.

But just to make sure there are no other possible values. For example, n=7: total sum is 5*180=900. Then, sum of all but one is 900 - x = 1070 => x = 900 - 1070 = -170, which is invalid. n=8 is valid. n=9 is invalid. n=10: total sum is 8*180=1440. Then, missing angle would be 1440 - 1070 = 370 degrees, which is way more than 180, invalid. So, no. Therefore, only n=8 is valid.

Therefore, the answer is 8 sides.

**Final Answer**
The convex polygon has \boxed{8} sides.

To determine the number of sides in a convex polygon where the sum of all but one of its interior angles is $1070^\circ$, we begin by recalling the formula for the sum of the interior angles of a convex polygon with $n$ sides:

$$
\text{Sum of interior angles} = (n - 2) \times 180^\circ
$$

Let the missing interior angle be $x$. Then, the sum of all but one of the interior angles is:

$$
(n - 2) \times 180 - x = 1070
$$

From this, we can solve for $x$:

$$
x = (n - 2) \times 180 - 1070
$$

Since the polygon is convex, all interior angles must be **less than $180^\circ$**. Therefore, we require:

$$
0 < x < 180
$$

Substituting the expression for $x$:

$$
0 < (n - 2) \times 180 - 1070 < 180
$$

---

### Step 1: Solve the Inequalities

**First inequality:**

$$
(n - 2) \times 180 - 1070 > 0
$$
$$
(n - 2) \times 180 > 1070
$$
$$
n - 2 > \frac{1070}{180} \approx 5.944
$$
$$
n > 7.944
$$

**Second inequality:**

$$
(n - 2) \times 180 - 1070 < 180
$$
$$
(n - 2) \times 180 < 1250
$$
$$
n - 2 < \frac{1250}{180} \approx 6.944
$$
$$
n < 8.944
$$

Combining both inequalities:

$$
7.944 < n < 8.944
$$

Since $n$ must be an integer, the only possible value is:

$$
n = 8
$$

---

### Step 2: Verify the Solution

For $n = 8$, the total sum of interior angles is:

$$
(8 - 2) \times 180 = 6 \times 180 = 1080^\circ
$$

If the sum of all but one angle is $1070^\circ$, then the missing angle is:

$$
x = 1080 - 1070 = 10^\circ
$$

Since $10^\circ < 180^\circ$, this is a valid interior angle for a convex polygon.

---

### Step 3: Check for Other Possible Values

- For $n = 7$: Total sum = $5 \times 180 = 900^\circ$, so missing angle = $900 - 1070 = -170^\circ$ (invalid).
- For $n = 9$: Total sum = $7 \times 180 = 1260^\circ$, so missing angle = $1260 - 1070 = 190^\circ$ (invalid, as it exceeds $180^\circ$).

Thus, only $n = 8$ satisfies all conditions.

---

### Final Answer

$$
\boxed{8}
$$

Solution:
The sum of the interior angles in any $n$-sided polygon is $180(n-2)$ degrees, so the angle measures in a polygon with 7 sides sum to $180(7-2) = 900$ degrees, which means that the desired polygon has more than 7 sides.  Meanwhile, the angle measures in a polygon with 8 sides sum to $180(8-2) = 1080$ degrees.  So, it's possible that the polygon has $\boxed{8}$ sides, and that the last angle measures $10^\circ$.

To see that this is the only possibility, note that the angle measures in a polygon with 9 sides sum to $180(9-2) = 1260$ degrees.  Therefore, if the polygon has more than 8 sides, then the last interior angle must measure at least $1260^\circ - 1070^\circ = 190^\circ$.  But this is impossible because each interior angle of a convex polygon has measure less than $180^\circ$.


Explanation:
Okay, so there's this problem about six cars pulling up to a red light one at a time. The light has three lanes: left-turn, straight-going, and right-turn. The question is asking how many ways the cars can stack up so that all three lanes are occupied. The cars are distinguishable, meaning each one is unique, and they come in a fixed order. So, the first car is different from the second, etc.

First, let me make sure I understand the problem correctly. We have six cars, each one arriving one after another. Each car can choose to go into one of the three lanes: left, straight, or right. However, we need to ensure that all three lanes are occupied. That means, after all six cars have arrived, each of the three lanes must have at least one car.

So, the problem is similar to assigning each car to one of three lanes, with the constraint that no lane is empty. But since the cars are distinguishable and the order in which they arrive matters, we need to consider permutations with constraints.

Wait, but actually, the cars are arriving one at a time, so the order in which they arrive is fixed. However, each car can choose which lane to go into. But the problem says "how many ways can the cars stack up so that all three lanes are occupied." So, maybe the stacking up refers to the order in which they are arranged in the lanes? Or is it about assigning each car to a lane, considering the order of arrival?

Wait, the note says that if the first car turns left and the second goes straight, that's different from the first car going straight and the second turning left. So, the order of the cars matters. So, each car is distinguishable, and their order of arrival is fixed. But each car can choose a lane. However, the problem is about the number of ways the cars can stack up so that all three lanes are occupied.

So, perhaps we need to count the number of assignments of the six cars to the three lanes, such that each lane has at least one car. However, since the cars are distinguishable and the order of arrival is fixed, but the assignment to lanes is what's being considered. Wait, but the problem says "stack up", so maybe the order in which they arrive affects the stacking? For example, if two cars are in the same lane, their order in the lane matters? Or is the stacking just about which lane each car is in, regardless of their order?

Wait, the note says that if the first car turns left and the second goes straight, that's different from the first car going straight and the second turning left. So, the order of the cars matters. So, perhaps the problem is about assigning each car to a lane, considering the order of arrival. But since the cars are arriving one at a time, each car's choice of lane is a decision that affects the overall configuration. However, the problem is about the number of ways the cars can stack up, meaning the number of possible sequences of lane choices such that all three lanes are occupied.

Alternatively, maybe the problem is about arranging the cars in the lanes, considering the order in which they arrive. For example, if two cars are in the same lane, their order in the lane is determined by their arrival order. So, the total number of ways would be the number of ways to assign each car to a lane (with the constraint that all three lanes are non-empty) multiplied by the number of ways to order the cars within each lane? Wait, but the problem says that the cars are distinguishable and pull up in a fixed order. So, maybe the order of arrival is fixed, and each car is assigned to a lane. Then, the stacking up refers to the assignment of lanes. However, the problem says "how many ways can the cars stack up so that all three lanes are occupied."

Wait, maybe the problem is similar to assigning each car to a lane, with the constraint that each lane has at least one car. Since the cars are distinguishable, the number of assignments would be the number of onto functions from the set of cars to the set of lanes. However, since the order of arrival is fixed, but the assignment to lanes is what's being considered. However, if the order of arrival is fixed, but the assignment to lanes is arbitrary, then the total number of assignments is 3^6, since each car can choose one of three lanes. But we need to subtract those assignments where one or more lanes are empty.

But the problem says "stack up", so maybe the order in which the cars are in each lane matters? For example, if two cars are in the left-turn lane, their order in the lane is determined by their arrival order. Therefore, the total number of ways would be the number of ways to assign each car to a lane (with all three lanes occupied) multiplied by the number of orderings within each lane? Wait, but if the order of arrival is fixed, then the order in which cars are in each lane is determined by their arrival order. Therefore, if we assign each car to a lane, the stacking up is determined by the assignment.

Wait, maybe the problem is equivalent to counting the number of functions from the six cars to the three lanes, such that each lane has at least one car. Since each car is assigned to a lane, and the order of arrival is fixed, so the assignment is just the mapping of each car to a lane. However, the problem says "stack up", so maybe the order in which they are in the lane matters? For example, if two cars are in the same lane, their order in the lane is determined by their arrival order. Therefore, the total number of ways would be the number of such assignments multiplied by the number of permutations within each lane? Wait, but if the order is determined by arrival, then the assignment is unique once you assign each car to a lane.

Wait, maybe the problem is not about assigning to lanes, but about arranging the cars in the lanes, considering the order. For example, if you have multiple cars in a lane, their order in the lane is important. However, since the cars arrive one at a time, the order in the lane is the same as the order of arrival. Therefore, if you have, say, two cars in the left-turn lane, the first car that went left is first in the lane, and the second car that went left is second in the lane. Therefore, the entire configuration is determined by which cars are assigned to which lane.

Therefore, the problem reduces to counting the number of ways to assign six distinguishable cars to three distinguishable lanes, with each lane having at least one car.

In that case, the answer would be the number of onto functions from the set of cars to the set of lanes, which is calculated using inclusion-exclusion.

The formula for the number of onto functions from a set of size n to a set of size k is:

k^n - C(k,1)*(k-1)^n + C(k,2)*(k-2)^n - ... + (-1)^{k}C(k,k)*(k - k)^n}

In this case, n=6, k=3. Therefore, the number is:

3^6 - C(3,1)*2^6 + C(3,2)*1^6 - C(3,3)*0^6

Calculating that:

3^6 = 729

C(3,1)*2^6 = 3*64 = 192

C(3,2)*1^6 = 3*1 = 3

C(3,3)*0^6 = 1*0 = 0

Therefore, the total is 729 - 192 + 3 - 0 = 729 - 192 is 537, plus 3 is 540.

So, 540 ways.

But wait, let me check if that's correct.

Alternatively, the number of onto functions is also given by the Stirling numbers of the second kind multiplied by k!.

The formula is:

Number of onto functions = S(n,k) * k!

Where S(n,k) is the Stirling number of the second kind.

For n=6, k=3, S(6,3) is 90. Then, multiplying by 3! = 6, gives 90*6=540.

Yes, that matches.

Therefore, the answer is 540.

But let me think again if this is the correct interpretation.

The problem says "how many ways can the cars stack up so that all three lanes are occupied."

If "stack up" refers to the order in which the cars are arranged in the lanes, considering that cars in the same lane are ordered by their arrival. Then, the total number of ways would be the same as assigning each car to a lane (with all lanes non-empty) and considering the order of arrival.

But since the order of arrival is fixed, once you assign cars to lanes, the order within each lane is fixed. Therefore, the total number of ways is indeed the number of onto functions, which is 540.

But let me think again. Suppose we have three lanes: left, straight, right. Each car can go into one of the three lanes. The cars are distinguishable, and the order in which they arrive is fixed. So, the problem is equivalent to assigning each car to a lane, with the constraint that all three lanes are non-empty.

Therefore, the answer is 3^6 - 3*2^6 + 3*1^6 = 540.

Alternatively, if the problem had considered the order within the lanes as part of the stacking, but since the order is determined by arrival, we don't need to multiply by anything else.

But let me check with a smaller case. Suppose there are 3 cars and 3 lanes. Then, the number of ways should be 3! = 6. According to the formula, it would be 3^3 - 3*2^3 + 3*1^3 = 27 - 24 + 3 = 6. Which is correct.

Another example: 4 cars, 3 lanes. The number of onto functions is 3^4 - 3*2^4 + 3*1^4 = 81 - 48 + 3 = 36. Alternatively, using Stirling numbers: S(4,3) = 6, multiplied by 3! = 6*6=36. Which is correct.

So, for 6 cars, 3 lanes, the answer is 540.

But let me check if there's another interpretation.

Suppose that the problem is about arranging the cars in the lanes, where the order in which they arrive affects the stacking. For example, if two cars are in the same lane, their order in the lane is determined by their arrival order. Therefore, the total number of ways is the number of ways to assign each car to a lane (with all lanes non-empty) multiplied by the number of permutations within each lane? Wait, but if the order is determined by arrival, then the assignment already determines the order.

Wait, suppose we have two cars in the left lane. The first car that went left is first in the lane, the second car that went left is second in the lane. Therefore, the order in the lane is fixed by the arrival order. Therefore, once you assign the cars to the lanes, the entire configuration is determined. Therefore, the number of configurations is equal to the number of assignments, which is the number of onto functions.

Therefore, the answer is 540.

But let me think again. Suppose that the problem is about the order in which the cars are in the lanes. For example, if you have multiple cars in a lane, their order in the lane is considered. However, since the cars arrive one at a time, the order in the lane is the same as the order of arrival. Therefore, the entire configuration is determined by which cars are in which lane.

Therefore, the answer is indeed 540.

Alternatively, if the problem had said that the order in the lane is arbitrary, then we would have to consider permutations. But since the cars arrive in a fixed order, the order in the lane is fixed.

Therefore, I think the answer is 540.

But let me check again with another approach.

Suppose we think of the problem as first assigning each car to a lane, then ensuring that all lanes are occupied.

Total number of assignments without restriction: 3^6 = 729.

Now, subtract the assignments where at least one lane is empty.

Number of assignments where left lane is empty: 2^6. Similarly for straight and right. So, subtract 3*2^6.

But then, we have subtracted too much. The cases where two lanes are empty have been subtracted multiple times. So, we need to add them back.

Number of assignments where left and straight are empty: 1^6. Similarly for other pairs. There are C(3,2)=3 such pairs.

So, add back 3*1^6.

Then, subtract the cases where all three lanes are empty, but that's impossible, so 0.

Therefore, total is 3^6 - 3*2^6 + 3*1^6 = 729 - 192 + 3 = 540.

Same result.

Therefore, I think the answer is 540.

But let me check if there's a different interpretation.

Suppose that the problem is about the order of the cars in the intersection. For example, if you have multiple cars in a lane, they are queued up in the order they arrived. So, the entire configuration is a sequence of cars, each assigned to a lane, with the constraint that all three lanes are non-empty.

But since the cars arrive one at a time, the order of the cars is fixed. However, the assignment to lanes is what's variable.

Wait, but if the order of arrival is fixed, then the entire configuration is determined by the assignment of each car to a lane. For example, if you have car 1, car 2, ..., car 6 arriving in that order, and each car is assigned to a lane, then the "stack up" is the sequence of lane assignments.

Therefore, the number of such sequences where all three lanes are used is 3^6 minus the ones that don't use all three lanes, which is 540.

Therefore, I think that is correct.

Alternatively, if the problem had said that the order in the lanes matters in some other way, but given the note, which says that the order of the cars matters (i.e., first car left, second straight is different from first straight, second left), then the assignment is just the sequence of lane choices.

Therefore, the answer is 540.

But let me check with another approach.

Suppose we think of the problem as first choosing which cars go into each lane, then arranging them.

But since the order of arrival is fixed, once we choose which cars go into each lane, the order in each lane is fixed.

Therefore, the number of ways is equal to the number of ways to partition the six cars into three non-empty subsets (one for each lane), multiplied by the number of ways to assign these subsets to the three lanes.

The number of ways to partition six cars into three non-empty subsets is the Stirling numbers of the second kind, S(6,3) = 90. Then, since the lanes are distinguishable (left, straight, right), we multiply by 3! = 6, giving 90*6=540.

Same result.

Therefore, this confirms the answer is 540.

Therefore, I think the answer is 540.

**Final Answer**
The number of ways the cars can stack up so that all three lanes are occupied is \boxed{540}.

To solve the problem, we are given six distinguishable cars arriving one at a time at a red light with three lanes: **left-turn**, **straight-going**, and **right-turn**. The goal is to determine how many **distinct ways** the cars can **stack up** such that **all three lanes are occupied**.

---

### Key Observations:

- **Distinguishable Cars**: Each car is unique, so the order in which they arrive matters.
- **Fixed Arrival Order**: The cars arrive in a fixed sequence (e.g., car 1, car 2, ..., car 6), so the order of arrival is not variable.
- **Lane Assignment**: Each car independently chooses one of the three lanes.
- **Lane Occupancy Constraint**: All three lanes must be **non-empty** after all six cars have arrived.

---

### Reformulating the Problem:

We are essentially assigning each of the six cars to one of the three lanes, with the constraint that **each lane must have at least one car**. Since the cars are distinguishable and the order of arrival is fixed, the **assignment of cars to lanes** determines the final configuration.

This is equivalent to counting the number of **onto functions** from a set of 6 elements (cars) to a set of 3 elements (lanes), where each lane is assigned at least one car.

---

### Counting the Valid Assignments:

We can use the **principle of inclusion-exclusion** to count the number of such assignments.

Let’s denote:

- $ n = 6 $ (number of cars)
- $ k = 3 $ (number of lanes)

The total number of functions from 6 cars to 3 lanes is $ 3^6 = 729 $.

Now, subtract the cases where at least one lane is empty:

- Number of functions where **one specific lane is empty**: $ 2^6 = 64 $
- There are $ \binom{3}{1} = 3 $ such cases (one for each lane being empty)

Now, add back the cases where **two lanes are empty** (i.e., all cars go to one lane):

- Number of functions where **two specific lanes are empty**: $ 1^6 = 1 $
- There are $ \binom{3}{2} = 3 $ such cases

Finally, subtract the case where **all three lanes are empty**, which is impossible (since all cars must be assigned to some lane), so this term is 0.

Putting it all together:

$$
\text{Valid assignments} = 3^6 - \binom{3}{1} \cdot 2^6 + \binom{3}{2} \cdot 1^6 - \binom{3}{3} \cdot 0^6
$$

$$
= 729 - 3 \cdot 64 + 3 \cdot 1 - 0 = 729 - 192 + 3 = 540
$$

---

### Alternative Interpretation via Stirling Numbers:

We can also think of this as partitioning the 6 cars into 3 **non-empty** subsets (one for each lane), and then assigning these subsets to the 3 lanes.

- The number of ways to partition 6 distinguishable items into 3 **non-empty** subsets is the **Stirling number of the second kind**, $ S(6, 3) = 90 $.
- Since the lanes are **distinguishable**, we multiply by the number of permutations of the 3 subsets, which is $ 3! = 6 $.

$$
\text{Total ways} = S(6, 3) \cdot 3! = 90 \cdot 6 = 540
$$

---

### Final Answer:

$$
\boxed{540}
$$

[Assistant]
Explanation:
Okay, so there's this problem about six cars pulling up to a red light one at a time. The light has three lanes: left-turn, straight-going, and right-turn. The question is asking how many ways the cars can stack up so that all three lanes are occupied. The cars are distinguishable, meaning each one is unique, and they come in a fixed order. So, the first car is different from the second, etc.

First, let me make sure I understand the problem correctly. We have six cars, each one arriving one after another. Each car can choose to go into one of the three lanes: left, straight, or right. However, we need to ensure that all three lanes are occupied. That means, after all six cars have arrived, each of the three lanes must have at least one car.

So, the problem is similar to assigning each car to one of three lanes, with the constraint that no lane is empty. But since the cars are distinguishable and the order in which they arrive matters, we need to consider permutations with constraints.

Wait, but actually, the cars are arriving one at a time, so the order in which they arrive is fixed. However, each car can choose which lane to go into. But the problem says "how many ways can the cars stack up so that all three lanes are occupied." So, maybe the stacking up refers to the order in which they are arranged in the lanes? Or is it about assigning each car to a lane, considering the order of arrival?

Wait, the note says that if the first car turns left and the second goes straight, that's different from the first car going straight and the second turning left. So, the order of the cars matters. So, each car is distinguishable, and their order of arrival is fixed. But each car can choose a lane. However, the problem is about the number of ways the cars can stack up so that all three lanes are occupied.

So, perhaps we need to count the number of assignments of the six cars to the three lanes, such that each lane has at least one car. However, since the cars are distinguishable and the order of arrival is fixed, but the assignment to lanes is what's being considered. Wait, but the problem says "stack up", so maybe the order in which they arrive affects the stacking? For example, if two cars are in the same lane, their order in the lane matters? Or is the stacking just about which lane each car is in, regardless of their order?

Wait, the note says that if the first car turns left and the second goes straight, that's different from the first car going straight and the second turning left. So, the order of the cars matters. So, perhaps the problem is about assigning each car to a lane, considering the order of arrival. But since the cars are arriving one at a time, each car's choice of lane is a decision that affects the overall configuration. However, the problem is about the number of ways the cars can stack up, meaning the number of possible sequences of lane choices such that all three lanes are occupied.

Alternatively, maybe the problem is about arranging the cars in the lanes, considering the order in which they arrive. For example, if two cars are in the same lane, their order in the lane is determined by their arrival order. So, the total number of ways would be the number of ways to assign each car to a lane (with the constraint that all three lanes are non-empty) multiplied by the number of ways to order the cars within each lane? Wait, but the problem says that the cars are distinguishable and pull up in a fixed order. So, maybe the order of arrival is fixed, and each car is assigned to a lane. Then, the stacking up refers to the assignment of lanes. However, the problem says "how many ways can the cars stack up so that all three lanes are occupied."

Wait, maybe the problem is similar to assigning each car to a lane, with the constraint that each lane has at least one car. Since the cars are distinguishable, the number of assignments would be the number of onto functions from the set of cars to the set of lanes. However, since the order of arrival is fixed, but the assignment to lanes is what's being considered. However, if the order of arrival is fixed, but the assignment to lanes is arbitrary, then the total number of assignments is 3^6, since each car can choose one of three lanes. But we need to subtract those assignments where one or more lanes are empty.

But the problem says "stack up", so maybe the order in which the cars are in each lane matters? For example, if two cars are in the left-turn lane, their order in the lane is determined by their arrival order. Therefore, the total number of ways would be the number of ways to assign each car to a lane (with all three lanes occupied) multiplied by the number of orderings within each lane? Wait, but if the order of arrival is fixed, then the order in which cars are in each lane is determined by their arrival order. Therefore, if we assign each car to a lane, the stacking up is determined by the assignment.

Wait, maybe the problem is equivalent to counting the number of functions from the six cars to the three lanes, such that each lane has at least one car. Since each car is assigned to a lane, and the order of arrival is fixed, so the assignment is just the mapping of each car to a lane. However, the problem says "stack up", so maybe the order in which they are in the lane matters? For example, if two cars are in the same lane, their order in the lane is determined by their arrival order. Therefore, the total number of ways would be the number of such assignments multiplied by the number of permutations within each lane? Wait, but if the order is determined by arrival, then the assignment is unique once you assign each car to a lane.

Wait, maybe the problem is not about assigning to lanes, but about arranging the cars in the lanes, considering the order. For example, if you have multiple cars in a lane, their order in the lane is important. However, since the cars arrive one at a time, the order in the lane is the same as the order of arrival. Therefore, if you have, say, two cars in the left-turn lane, the first car that went left is first in the lane, and the second car that went left is second in the lane. Therefore, the entire configuration is determined by which cars are assigned to which lane.

Therefore, the problem reduces to counting the number of ways to assign six distinguishable cars to three distinguishable lanes, with each lane having at least one car.

In that case, the answer would be the number of onto functions from the set of cars to the set of lanes, which is calculated using inclusion-exclusion.

The formula for the number of onto functions from a set of size n to a set of size k is:

k^n - C(k,1)*(k-1)^n + C(k,2)*(k-2)^n - ... + (-1)^{k}C(k,k)*(k - k)^n}

In this case, n=6, k=3. Therefore, the number is:

3^6 - C(3,1)*2^6 + C(3,2)*1^6 - C(3,3)*0^6

Calculating that:

3^6 = 729

C(3,1)*2^6 = 3*64 = 192

C(3,2)*1^6 = 3*1 = 3

C(3,3)*0^6 = 1*0 = 0

Therefore, the total is 729 - 192 + 3 - 0 = 729 - 192 is 537, plus 3 is 540.

So, 540 ways.

But wait, let me check if that's correct.

Alternatively, the number of onto functions is also given by the Stirling numbers of the second kind multiplied by k!.

The formula is:

Number of onto functions = S(n,k) * k!

Where S(n,k) is the Stirling number of the second kind.

For n=6, k=3, S(6,3) is 90. Then, multiplying by 3! = 6, gives 90*6=540.

Yes, that matches.

Therefore, the answer is 540.

But let me think again if this is the correct interpretation.

The problem says "how many ways can the cars stack up so that all three lanes are occupied."

If "stack up" refers to the order in which the cars are arranged in the lanes, considering that cars in the same lane are ordered by their arrival. Then, the total number of ways would be the same as assigning each car to a lane (with all lanes non-empty) and considering the order of arrival.

But since the order of arrival is fixed, once you assign cars to lanes, the order within each lane is fixed. Therefore, the total number of ways is indeed the number of onto functions, which is 540.

But let me think again. Suppose we have three lanes: left, straight, right. Each car can go into one of the three lanes. The cars are distinguishable, and the order in which they arrive is fixed. So, the problem is equivalent to assigning each car to a lane, with the constraint that all three lanes are non-empty.

Therefore, the answer is 3^6 - 3*2^6 + 3*1^6 = 540.

Alternatively, if the problem had considered the order within the lanes as part of the stacking, but since the order is determined by arrival, we don't need to multiply by anything else.

But let me check with a smaller case. Suppose there are 3 cars and 3 lanes. Then, the number of ways should be 3! = 6. According to the formula, it would be 3^3 - 3*2^3 + 3*1^3 = 27 - 24 + 3 = 6. Which is correct.

Another example: 4 cars, 3 lanes. The number of onto functions is 3^4 - 3*2^4 + 3*1^4 = 81 - 48 + 3 = 36. Alternatively, using Stirling numbers: S(4,3) = 6, multiplied by 3! = 6*6=36. Which is correct.

So, for 6 cars, 3 lanes, the answer is 540.

But let me check if there's another interpretation.

Suppose that the problem is about arranging the cars in the lanes, where the order in which they arrive affects the stacking. For example, if two cars are in the same lane, their order in the lane is determined by their arrival order. Therefore, the total number of ways is the number of ways to assign each car to a lane (with all lanes non-empty) multiplied by the number of permutations within each lane? Wait, but if the order is determined by arrival, then the assignment already determines the order.

Wait, suppose we have two cars in the left lane. The first car that went left is first in the lane, the second car that went left is second in the lane. Therefore, the order in the lane is fixed by the arrival order. Therefore, once you assign the cars to the lanes, the entire configuration is determined. Therefore, the number of configurations is equal to the number of assignments, which is the number of onto functions.

Therefore, the answer is 540.

But let me think again. Suppose that the problem is about the order in which the cars are in the lanes. For example, if you have multiple cars in a lane, their order in the lane is considered. However, since the cars arrive one at a time, the order in the lane is the same as the order of arrival. Therefore, the entire configuration is determined by which cars are in which lane.

Therefore, the answer is indeed 540.

Alternatively, if the problem had said that the order in the lane is arbitrary, then we would have to consider permutations. But since the cars arrive in a fixed order, the order in the lane is fixed.

Therefore, I think the answer is 540.

But let me check again with another approach.

Suppose we think of the problem as first assigning each car to a lane, then ensuring that all lanes are occupied.

Total number of assignments without restriction: 3^6 = 729.

Now, subtract the assignments where at least one lane is empty.

Number of assignments where left lane is empty: 2^6. Similarly for straight and right. So, subtract 3*2^6.

But then, we have subtracted too much. The cases where two lanes are empty have been subtracted multiple times. So, we need to add them back.

Number of assignments where left and straight are empty: 1^6. Similarly for other pairs. There are C(3,2)=3 such pairs.

So, add back 3*1^6.

Then, subtract the cases where all three lanes are empty, but that's impossible, so 0.

Therefore, total is 3^6 - 3*2^6 + 3*1^6 = 729 - 192 + 3 = 540.

Same result.

Therefore, I think the answer is 540.

But let me check if there's a different interpretation.

Suppose that the problem is about the order of the cars in the intersection. For example, if you have multiple cars in a lane, they are queued up in the order they arrived. So, the entire configuration is a sequence of cars, each assigned to a lane, with the constraint that all three lanes are non-empty.

But since the cars arrive one at a time, the order of the cars is fixed. However, the assignment to lanes is what's variable.

Wait, but if the order of arrival is fixed, then the entire configuration is determined by the assignment of each car to a lane. For example, if you have car 1, car 2, ..., car 6 arriving in that order, and each car is assigned to a lane, then the "stack up" is the sequence of lane assignments.

Therefore, the number of such sequences where all three lanes are used is 3^6 minus the ones that don't use all three lanes, which is 540.

Therefore, I think that is correct.

Alternatively, if the problem had said that the order in the lanes matters in some other way, but given the note, which says that the order of the cars matters (i.e., first car left, second straight is different from first straight, second left), then the assignment is just the sequence of lane choices.

Therefore, the answer is 540.

But let me check with another approach.

Suppose we think of the problem as first choosing which cars go into each lane, then arranging them.

But since the order of arrival is fixed, once we choose which cars go into each lane, the order in each lane is fixed.

Therefore, the number of ways is equal to the number of ways to partition the six cars into three non-empty subsets (one for each lane), multiplied by the number of ways to assign these subsets to the three lanes.

The number of ways to partition six cars into three non-empty subsets is the Stirling numbers of the second kind, S(6,3) = 90. Then, since the lanes are distinguishable (left, straight, right), we multiply by 3! = 6, giving 90*6=540.

Same result.

Therefore, this confirms the answer is 540.

Therefore, I think the answer is 540.

**Final Answer**
The number of ways the cars can stack up so that all three lanes are occupied is \boxed{540}.

To solve the problem, we are given six distinguishable cars arriving one at a time at a red light with three lanes: **left-turn**, **straight-going**, and **right-turn**. The goal is to determine how many **distinct ways** the cars can **stack up** such that **all three lanes are occupied**.

---

### Key Observations:

- **Distinguishable Cars**: Each car is unique, so the order in which they arrive matters.
- **Fixed Arrival Order**: The cars arrive in a fixed sequence (e.g., car 1, car 2, ..., car 6), so the order of arrival is not variable.
- **Lane Assignment**: Each car independently chooses one of the three lanes.
- **Lane Occupancy Constraint**: All three lanes must be **non-empty** after all six cars have arrived.

---

### Reformulating the Problem:

We are essentially assigning each of the six cars to one of the three lanes, with the constraint that **each lane must have at least one car**. Since the cars are distinguishable and the order of arrival is fixed, the **assignment of cars to lanes** determines the final configuration.

This is equivalent to counting the number of **onto functions** from a set of 6 elements (cars) to a set of 3 elements (lanes), where each lane is assigned at least one car.

---

### Counting the Valid Assignments:

We can use the **principle of inclusion-exclusion** to count the number of such assignments.

Let’s denote:

- $ n = 6 $ (number of cars)
- $ k = 3 $ (number of lanes)

The total number of functions from 6 cars to 3 lanes is $ 3^6 = 729 $.

Now, subtract the cases where at least one lane is empty:

- Number of functions where **one specific lane is empty**: $ 2^6 = 64 $
- There are $ \binom{3}{1} = 3 $ such cases (one for each lane being empty)

Now, add back the cases where **two lanes are empty** (i.e., all cars go to one lane):

- Number of functions where **two specific lanes are empty**: $ 1^6 = 1 $
- There are $ \binom{3}{2} = 3 $ such cases

Finally, subtract the case where **all three lanes are empty**, which is impossible (since all cars must be assigned to some lane), so this term is 0.

Putting it all together:

$$
\text{Valid assignments} = 3^6 - \binom{3}{1} \cdot 2^6 + \binom{3}{2} \cdot 1^6 - \binom{3}{3} \cdot 0^6
$$

$$
= 729 - 3 \cdot 64 + 3 \cdot 1 - 0 = 729 - 192 + 3 = 540
$$

---

### Alternative Interpretation via Stirling Numbers:

We can also think of this as partitioning the 6 cars into 3 **non-empty** subsets (one for each lane), and then assigning these subsets to the 3 lanes.

- The number of ways to partition 6 distinguishable items into 3 **non-empty** subsets is the **Stirling number of the second kind**, $ S(6, 3) = 90 $.
- Since the lanes are **distinguishable**, we multiply by the number of permutations of the 3 subsets, which is $ 3! = 6 $.

$$
\text{Total ways} = S(6, 3) \cdot 3! = 90 \cdot 6 = 540
$$

---

### Final Answer:

$$
\boxed{540}
$$

Solution: We count the number of ways that some lane can be left empty, and subtract from the total number, $3^6=729$ because each driver has three choices. Suppose the left-turn lane is left empty. Then each driver is limited to 2 choices, and there are $2^6$ ways to leave the left-turn lane empty. The same logic gives $2^6$ ways to leave the center lane and right-turn lane open. But we have double-counted the situations where two lanes are left empty. Fortunately, since each driver must go into the third lane, there are only 3 situations we have over-counted. This leaves $3\cdot2^6-3=189$ ways to leave at least one lane unoccupied, or $729-189=\boxed{540}$ ways to occupy every lane.


Explanation:
Okay, so I need to find the maximum value of the function $ 4(x + 7)(2 - x) $. Hmm, let me think about how to approach this. I remember that quadratic functions have either a maximum or a minimum value, depending on the coefficient of the $ x^2 $ term. Since the problem is asking for the maximum value, I should check if this is a quadratic function and then find its vertex.

First, let me expand the expression to make it easier to analyze. Let me multiply out the terms:

$ 4(x + 7)(2 - x) $

Let me first compute $ (x + 7)(2 - x) $. Let me use the distributive property (FOIL method):

Multiply the First terms: $ x \times 2 = 2x $

Outer terms: $ x \times (-x) = -x^2 $

Inner terms: $ 7 \times 2 = 14 $

Last terms: $ 7 \times (-x) = -7x $

So combining all these terms:

$ 2x - x^2 + 14 - 7x $

Combine like terms:

$ (2x - 7x) + (-x^2) + 14 $

Which is:

$ -5x - x^2 + 14 $

Let me rearrange the terms in standard quadratic form:

$ -x^2 - 5x + 14 $

Now, the entire original expression is multiplied by 4:

$ 4(-x^2 - 5x + 14) $

Let me distribute the 4:

$ -4x^2 - 20x + 56 $

So the function simplifies to $ f(x) = -4x^2 - 20x + 56 $.

Since the coefficient of $ x^2 $ is negative (-4), this parabola opens downward, which means it has a maximum value at its vertex. Therefore, the maximum value occurs at the vertex of this parabola.

Now, I need to find the vertex of the quadratic function. The vertex of a parabola given by $ f(x) = ax^2 + bx + c $ is at $ x = -\frac{b}{2a} $.

In this case, $ a = -4 $ and $ b = -20 $. Plugging into the formula:

$ x = -(-20)/(2 * -4) = 20 / (-8) = -2.5 $

Wait, let me check that again.

Wait, $ x = -b/(2a) $. So here, $ b = -20 $, so:

$ x = -(-20)/(2 * -4) = 20 / (-8) = -2.5 $. Yes, that's correct. So the x-coordinate of the vertex is -2.5.

Now, to find the maximum value, I need to plug this x-value back into the function. Let me compute $ f(-2.5) $.

Alternatively, maybe I can use the vertex formula for the maximum value. Wait, another way to find the maximum value is using the formula $ c - \frac{b^2}{4a} $, but I need to recall if that's correct. Wait, no, that might not be the case. Let me think.

Alternatively, since I have the quadratic in standard form, maybe I can complete the square. But maybe it's easier to just plug in the x-value. Let me try that.

First, let me write down the function again:

$ f(x) = -4x^2 - 20x + 56 $

So plugging in x = -2.5:

First, compute each term:

-4x²: -4*(-2.5)^2

-20x: -20*(-2.5)

+56: +56

Let me compute each term step by step.

First, (-2.5)^2 is 6.25. Then, -4 * 6.25 = -25.

Second term: -20*(-2.5) = +50.

Third term is +56.

So adding them up: -25 + 50 + 56 = (-25 + 50) + 56 = 25 + 56 = 81.

So the maximum value is 81? Wait, that seems straightforward. Let me check if I did that correctly.

Alternatively, maybe I can check by expanding the original expression. Let me try another approach. Let me go back to the original expression before expanding: $ 4(x + 7)(2 - x) $. Maybe I can find the maximum by completing the square or using calculus? Wait, but since this is a quadratic, calculus might be overkill, but let me try.

Alternatively, maybe I can use the vertex formula. Let me recall that for a quadratic function $ ax^2 + bx + c $, the maximum or minimum value is $ f(-b/(2a)) $. So here, we already found that x = -2.5, and plugging back in gives 81. Let me check with another method.

Alternatively, maybe I can use the formula for the vertex. Let me think. Alternatively, maybe I can write the original expression as a quadratic in terms of (x + something). Let me try completing the square.

Starting from the original expression: $ 4(x + 7)(2 - x) $. Let me write it as $ 4(-x^2 -5x +14) $, as before. Let me factor out the -4 from the quadratic part:

Wait, $ -4x^2 -20x +56 = -4(x^2 + 5x) + 56 $. Wait, but completing the square for $ x^2 + 5x $. Let me do that.

Take $ x^2 + 5x $. To complete the square, take half of 5, which is 2.5, square it, which is 6.25. So:

$ x^2 + 5x + 6.25 - 6.25 = (x + 2.5)^2 - 6.25 $

Therefore, substituting back into the expression:

$ -4[(x + 2.5)^2 - 6.25] + 56 $

Which is:

$ -4(x + 2.5)^2 + 25 + 56 $

Which is:

$ -4(x + 2.5)^2 + 81 $

Therefore, the function can be written as $ -4(x + 2.5)^2 + 81 $. Since the square term is always non-negative, multiplying by -4 makes it non-positive, so the maximum value occurs when the square term is zero, i.e., when $ x = -2.5 $, and then the maximum value is 81.

So that confirms the previous result. Therefore, the maximum value is 81.

Alternatively, maybe I can check with calculus. Taking the derivative of f(x) = -4x² -20x +56. The derivative is f’(x) = -8x -20. Setting derivative equal to zero: -8x -20 = 0 => -8x = 20 => x = -20/8 = -2.5. Then, plugging back into f(x) gives the same result as before. So that's consistent.

Alternatively, maybe I can check with another value. Let me pick x = -2.5 and compute the original expression. Let me compute 4(x + 7)(2 - x).

x = -2.5:

x + 7 = -2.5 + 7 = 4.5

2 - x = 2 - (-2.5) = 2 + 2.5 = 4.5

Therefore, 4*(4.5)*(4.5) = 4*(20.25) = 81. Yep, same result.

So that seems to confirm that the maximum value is indeed 81.

Alternatively, maybe I can think of this as a quadratic in terms of (x + 7) or something else? Let me see.

Alternatively, maybe I can use the concept that the maximum of a quadratic function occurs at the vertex, which we found.

Alternatively, maybe I can use the formula for the maximum of a quadratic function. Wait, but I think all methods lead to the same answer.

Wait, just to make sure, let me try plugging in some other x-values to see if the function is indeed lower. For example, let me try x = 0.

Original function: 4*(0 + 7)*(2 - 0) = 4*7*2 = 56. Which is less than 81.

How about x = -2.5? As we saw, it's 81.

How about x = -3? Let's compute:

Original expression: 4*(-3 + 7)*(2 - (-3)) = 4*(4)*(5) = 4*20 = 80. Which is less than 81.

How about x = -2?

Original expression: 4*(-2 + 7)*(2 - (-2)) = 4*(5)*(4) = 4*20 = 80. Also less than 81.

How about x = -2.5? As before, 81.

How about x = -2.5 + 1 = -1.5? Let me compute:

Original expression: 4*(-1.5 + 7)*(2 - (-1.5)) = 4*(5.5)*(3.5). Let me compute that.

First, 5.5 * 3.5. Let me compute 5 * 3.5 = 17.5, and 0.5 * 3.5 = 1.75, so total is 17.5 + 1.75 = 19.25. Then multiply by 4: 19.25 * 4 = 77. Which is less than 81.

Similarly, if I try x = -2.5 - 1 = -3.5:

Original expression: 4*(-3.5 + 7)*(2 - (-3.5)) = 4*(3.5)*(5.5). Which is same as above, 4*3.5*5.5 = same as before, 77.

So that seems to confirm that the maximum is indeed at x = -2.5, with value 81.

Alternatively, maybe I can think of the original expression as a product of two linear terms, and find its maximum. Let me think.

The original expression is 4(x + 7)(2 - x). Let me consider the function g(x) = (x + 7)(2 - x). Then, the original function is 4g(x). So, if I can find the maximum of g(x), then multiply by 4.

So, let me find the maximum of g(x) = (x + 7)(2 - x). Let me expand this:

g(x) = -x² -5x +14, same as before. So, same quadratic. So, same maximum.

Alternatively, maybe I can think of this as a quadratic in terms of x, and use the vertex formula.

Alternatively, maybe I can use the concept of symmetry. The product (x + 7)(2 - x) can be thought of as a quadratic function, which is symmetric around its vertex.

Alternatively, maybe I can use substitution. Let me set y = x + something.

But I think all methods lead to the same conclusion.

Alternatively, maybe I can use the AM-GM inequality? Wait, but AM-GM applies to positive numbers. However, here, the terms (x + 7) and (2 - x) might not be positive for all x. Let me check.

For example, if x is between -7 and 2, then both (x + 7) and (2 - x) are positive. If x < -7, then (x + 7) is negative and (2 - x) is positive (since x is negative, so -x is positive, so 2 - x is greater than 2). If x > 2, then (2 - x) is negative and (x + 7) is positive. So, the product is positive only when x is between -7 and 2. Outside of that interval, the product is negative. Therefore, the maximum value is achieved somewhere in the interval (-7, 2).

But since we already found the vertex at x = -2.5, which is within (-7, 2), that's the maximum.

Alternatively, if I wanted to use calculus, I could take the derivative, set it to zero, and find critical points. But since we already did that, and confirmed with multiple methods, I think we can be confident.

Alternatively, maybe I can use a graphing approach. If I imagine the graph of the function, it's a downward opening parabola with vertex at (-2.5, 81). So, the maximum value is 81.

Therefore, after all these methods and checks, I think the maximum value is indeed 81.

**Final Answer**
The maximum value is \boxed{81}.

To find the **maximum value** of the expression:

$$
4(x + 7)(2 - x)
$$

we begin by recognizing that this is a **quadratic expression** in terms of $ x $. The goal is to determine the **maximum value** of this function over all real numbers $ x $.

---

### Step 1: Expand the Expression

We first expand the product:

$$
(x + 7)(2 - x) = x(2 - x) + 7(2 - x) = 2x - x^2 + 14 - 7x = -x^2 - 5x + 14
$$

Now multiply by 4:

$$
4(x + 7)(2 - x) = 4(-x^2 - 5x + 14) = -4x^2 - 20x + 56
$$

So the function becomes:

$$
f(x) = -4x^2 - 20x + 56
$$

---

### Step 2: Identify the Type of Function

This is a **quadratic function** of the form:

$$
f(x) = ax^2 + bx + c
$$

with $ a = -4 $, $ b = -20 $, and $ c = 56 $.

Since the coefficient of $ x^2 $ is **negative**, the parabola opens **downward**, meaning the function has a **maximum** value at its **vertex**.

---

### Step 3: Find the Vertex

The x-coordinate of the vertex of a quadratic function is given by:

$$
x = -\frac{b}{2a}
$$

Substituting the values:

$$
x = -\frac{-20}{2(-4)} = \frac{20}{-8} = -2.5
$$

---

### Step 4: Evaluate the Function at the Vertex

Now plug $ x = -2.5 $ into the original expression:

$$
4(x + 7)(2 - x) = 4(-2.5 + 7)(2 - (-2.5)) = 4(4.5)(4.5) = 4 \cdot 20.25 = 81
$$

---

### Step 5: Confirm the Maximum

We can also verify this by completing the square or using calculus, but all methods consistently show that the **maximum value** of the function is achieved at $ x = -2.5 $, and the value is:

$$
\boxed{81}
$$

---

### Final Answer

$$
\boxed{81}
$$

[Assistant]
Explanation:
Okay, so I need to find the maximum value of the function $ 4(x + 7)(2 - x) $. Hmm, let me think about how to approach this. I remember that quadratic functions have either a maximum or a minimum value, depending on the coefficient of the $ x^2 $ term. Since the problem is asking for the maximum value, I should check if this is a quadratic function and then find its vertex.

First, let me expand the expression to make it easier to analyze. Let me multiply out the terms:

$ 4(x + 7)(2 - x) $

Let me first compute $ (x + 7)(2 - x) $. Let me use the distributive property (FOIL method):

Multiply the First terms: $ x \times 2 = 2x $

Outer terms: $ x \times (-x) = -x^2 $

Inner terms: $ 7 \times 2 = 14 $

Last terms: $ 7 \times (-x) = -7x $

So combining all these terms:

$ 2x - x^2 + 14 - 7x $

Combine like terms:

$ (2x - 7x) + (-x^2) + 14 $

Which is:

$ -5x - x^2 + 14 $

Let me rearrange the terms in standard quadratic form:

$ -x^2 - 5x + 14 $

Now, the entire original expression is multiplied by 4:

$ 4(-x^2 - 5x + 14) $

Let me distribute the 4:

$ -4x^2 - 20x + 56 $

So the function simplifies to $ f(x) = -4x^2 - 20x + 56 $.

Since the coefficient of $ x^2 $ is negative (-4), this parabola opens downward, which means it has a maximum value at its vertex. Therefore, the maximum value occurs at the vertex of this parabola.

Now, I need to find the vertex of the quadratic function. The vertex of a parabola given by $ f(x) = ax^2 + bx + c $ is at $ x = -\frac{b}{2a} $.

In this case, $ a = -4 $ and $ b = -20 $. Plugging into the formula:

$ x = -(-20)/(2 * -4) = 20 / (-8) = -2.5 $

Wait, let me check that again.

Wait, $ x = -b/(2a) $. So here, $ b = -20 $, so:

$ x = -(-20)/(2 * -4) = 20 / (-8) = -2.5 $. Yes, that's correct. So the x-coordinate of the vertex is -2.5.

Now, to find the maximum value, I need to plug this x-value back into the function. Let me compute $ f(-2.5) $.

Alternatively, maybe I can use the vertex formula for the maximum value. Wait, another way to find the maximum value is using the formula $ c - \frac{b^2}{4a} $, but I need to recall if that's correct. Wait, no, that might not be the case. Let me think.

Alternatively, since I have the quadratic in standard form, maybe I can complete the square. But maybe it's easier to just plug in the x-value. Let me try that.

First, let me write down the function again:

$ f(x) = -4x^2 - 20x + 56 $

So plugging in x = -2.5:

First, compute each term:

-4x²: -4*(-2.5)^2

-20x: -20*(-2.5)

+56: +56

Let me compute each term step by step.

First, (-2.5)^2 is 6.25. Then, -4 * 6.25 = -25.

Second term: -20*(-2.5) = +50.

Third term is +56.

So adding them up: -25 + 50 + 56 = (-25 + 50) + 56 = 25 + 56 = 81.

So the maximum value is 81? Wait, that seems straightforward. Let me check if I did that correctly.

Alternatively, maybe I can check by expanding the original expression. Let me try another approach. Let me go back to the original expression before expanding: $ 4(x + 7)(2 - x) $. Maybe I can find the maximum by completing the square or using calculus? Wait, but since this is a quadratic, calculus might be overkill, but let me try.

Alternatively, maybe I can use the vertex formula. Let me recall that for a quadratic function $ ax^2 + bx + c $, the maximum or minimum value is $ f(-b/(2a)) $. So here, we already found that x = -2.5, and plugging back in gives 81. Let me check with another method.

Alternatively, maybe I can use the formula for the vertex. Let me think. Alternatively, maybe I can write the original expression as a quadratic in terms of (x + something). Let me try completing the square.

Starting from the original expression: $ 4(x + 7)(2 - x) $. Let me write it as $ 4(-x^2 -5x +14) $, as before. Let me factor out the -4 from the quadratic part:

Wait, $ -4x^2 -20x +56 = -4(x^2 + 5x) + 56 $. Wait, but completing the square for $ x^2 + 5x $. Let me do that.

Take $ x^2 + 5x $. To complete the square, take half of 5, which is 2.5, square it, which is 6.25. So:

$ x^2 + 5x + 6.25 - 6.25 = (x + 2.5)^2 - 6.25 $

Therefore, substituting back into the expression:

$ -4[(x + 2.5)^2 - 6.25] + 56 $

Which is:

$ -4(x + 2.5)^2 + 25 + 56 $

Which is:

$ -4(x + 2.5)^2 + 81 $

Therefore, the function can be written as $ -4(x + 2.5)^2 + 81 $. Since the square term is always non-negative, multiplying by -4 makes it non-positive, so the maximum value occurs when the square term is zero, i.e., when $ x = -2.5 $, and then the maximum value is 81.

So that confirms the previous result. Therefore, the maximum value is 81.

Alternatively, maybe I can check with calculus. Taking the derivative of f(x) = -4x² -20x +56. The derivative is f’(x) = -8x -20. Setting derivative equal to zero: -8x -20 = 0 => -8x = 20 => x = -20/8 = -2.5. Then, plugging back into f(x) gives the same result as before. So that's consistent.

Alternatively, maybe I can check with another value. Let me pick x = -2.5 and compute the original expression. Let me compute 4(x + 7)(2 - x).

x = -2.5:

x + 7 = -2.5 + 7 = 4.5

2 - x = 2 - (-2.5) = 2 + 2.5 = 4.5

Therefore, 4*(4.5)*(4.5) = 4*(20.25) = 81. Yep, same result.

So that seems to confirm that the maximum value is indeed 81.

Alternatively, maybe I can think of this as a quadratic in terms of (x + 7) or something else? Let me see.

Alternatively, maybe I can use the concept that the maximum of a quadratic function occurs at the vertex, which we found.

Alternatively, maybe I can use the formula for the maximum of a quadratic function. Wait, but I think all methods lead to the same answer.

Wait, just to make sure, let me try plugging in some other x-values to see if the function is indeed lower. For example, let me try x = 0.

Original function: 4*(0 + 7)*(2 - 0) = 4*7*2 = 56. Which is less than 81.

How about x = -2.5? As we saw, it's 81.

How about x = -3? Let's compute:

Original expression: 4*(-3 + 7)*(2 - (-3)) = 4*(4)*(5) = 4*20 = 80. Which is less than 81.

How about x = -2?

Original expression: 4*(-2 + 7)*(2 - (-2)) = 4*(5)*(4) = 4*20 = 80. Also less than 81.

How about x = -2.5? As before, 81.

How about x = -2.5 + 1 = -1.5? Let me compute:

Original expression: 4*(-1.5 + 7)*(2 - (-1.5)) = 4*(5.5)*(3.5). Let me compute that.

First, 5.5 * 3.5. Let me compute 5 * 3.5 = 17.5, and 0.5 * 3.5 = 1.75, so total is 17.5 + 1.75 = 19.25. Then multiply by 4: 19.25 * 4 = 77. Which is less than 81.

Similarly, if I try x = -2.5 - 1 = -3.5:

Original expression: 4*(-3.5 + 7)*(2 - (-3.5)) = 4*(3.5)*(5.5). Which is same as above, 4*3.5*5.5 = same as before, 77.

So that seems to confirm that the maximum is indeed at x = -2.5, with value 81.

Alternatively, maybe I can think of the original expression as a product of two linear terms, and find its maximum. Let me think.

The original expression is 4(x + 7)(2 - x). Let me consider the function g(x) = (x + 7)(2 - x). Then, the original function is 4g(x). So, if I can find the maximum of g(x), then multiply by 4.

So, let me find the maximum of g(x) = (x + 7)(2 - x). Let me expand this:

g(x) = -x² -5x +14, same as before. So, same quadratic. So, same maximum.

Alternatively, maybe I can think of this as a quadratic in terms of x, and use the vertex formula.

Alternatively, maybe I can use the concept of symmetry. The product (x + 7)(2 - x) can be thought of as a quadratic function, which is symmetric around its vertex.

Alternatively, maybe I can use substitution. Let me set y = x + something.

But I think all methods lead to the same conclusion.

Alternatively, maybe I can use the AM-GM inequality? Wait, but AM-GM applies to positive numbers. However, here, the terms (x + 7) and (2 - x) might not be positive for all x. Let me check.

For example, if x is between -7 and 2, then both (x + 7) and (2 - x) are positive. If x < -7, then (x + 7) is negative and (2 - x) is positive (since x is negative, so -x is positive, so 2 - x is greater than 2). If x > 2, then (2 - x) is negative and (x + 7) is positive. So, the product is positive only when x is between -7 and 2. Outside of that interval, the product is negative. Therefore, the maximum value is achieved somewhere in the interval (-7, 2).

But since we already found the vertex at x = -2.5, which is within (-7, 2), that's the maximum.

Alternatively, if I wanted to use calculus, I could take the derivative, set it to zero, and find critical points. But since we already did that, and confirmed with multiple methods, I think we can be confident.

Alternatively, maybe I can use a graphing approach. If I imagine the graph of the function, it's a downward opening parabola with vertex at (-2.5, 81). So, the maximum value is 81.

Therefore, after all these methods and checks, I think the maximum value is indeed 81.

**Final Answer**
The maximum value is \boxed{81}.

To find the **maximum value** of the expression:

$$
4(x + 7)(2 - x)
$$

we begin by recognizing that this is a **quadratic expression** in terms of $ x $. The goal is to determine the **maximum value** of this function over all real numbers $ x $.

---

### Step 1: Expand the Expression

We first expand the product:

$$
(x + 7)(2 - x) = x(2 - x) + 7(2 - x) = 2x - x^2 + 14 - 7x = -x^2 - 5x + 14
$$

Now multiply by 4:

$$
4(x + 7)(2 - x) = 4(-x^2 - 5x + 14) = -4x^2 - 20x + 56
$$

So the function becomes:

$$
f(x) = -4x^2 - 20x + 56
$$

---

### Step 2: Identify the Type of Function

This is a **quadratic function** of the form:

$$
f(x) = ax^2 + bx + c
$$

with $ a = -4 $, $ b = -20 $, and $ c = 56 $.

Since the coefficient of $ x^2 $ is **negative**, the parabola opens **downward**, meaning the function has a **maximum** value at its **vertex**.

---

### Step 3: Find the Vertex

The x-coordinate of the vertex of a quadratic function is given by:

$$
x = -\frac{b}{2a}
$$

Substituting the values:

$$
x = -\frac{-20}{2(-4)} = \frac{20}{-8} = -2.5
$$

---

### Step 4: Evaluate the Function at the Vertex

Now plug $ x = -2.5 $ into the original expression:

$$
4(x + 7)(2 - x) = 4(-2.5 + 7)(2 - (-2.5)) = 4(4.5)(4.5) = 4 \cdot 20.25 = 81
$$

---

### Step 5: Confirm the Maximum

We can also verify this by completing the square or using calculus, but all methods consistently show that the **maximum value** of the function is achieved at $ x = -2.5 $, and the value is:

$$
\boxed{81}
$$

---

### Final Answer

$$
\boxed{81}
$$

Solution:
The graph of $y = 4(x + 7)(2 - x)$ is a parabola.  Since $y = 0$ when $x = -7$ and $x = 2$, the $x$-intercepts of the parabola are $(-7,0)$ and $(2,0)$.  If the vertex of the parabola is $(h,k)$, then the $x$-intercepts $(-7,0)$ and $(2,0)$ are symmetric around the line $x = h$, so $h = (-7 + 2)/2 = -5/2$.

Hence, the maximum value of $y = 4(x + 7)(2 - x)$ occurs at $x = -5/2$, in which case \[y = 4 \left( -\frac{5}{2} + 7 \right) \left( 2 + \frac{5}{2} \right) = 4 \cdot \frac{9}{2} \cdot \frac{9}{2} = \boxed{81}.\] (Note that this is a maximum value, and not a minimum value, because the coefficient of $x^2$ in $y = 4(x + 7)(2 - x) = -4x^2 - 20x + 56$ is negative.)