Explanation:
Okay, so I need to figure out how many cylindrical beakers it takes to fill a spherical tank. Let me start by recalling the formulas for the volumes of a cylinder and a sphere.

The volume of a cylinder is given by V_cylinder = πr²h, right? And the volume of a sphere is V_sphere = (4/3)πr³.

Alright, the problem says the beaker is 8 cm high and has a radius of 3 cm. So, first, I should calculate the volume of one beaker. Let me write that down:

V_beaker = π * (3 cm)² * 8 cm. Let me compute that.

First, 3 squared is 9, so that's 9 cm². Then multiply by 8 cm, so 9 * 8 = 72 cm³. Then multiply by π, so the volume of one beaker is 72π cm³.

Now, the spherical tank has a radius of 6 cm. Let me calculate its volume.

V_sphere = (4/3)π*(6 cm)³. Let me compute 6 cubed first. 6*6=36, 36*6=216. So that's 216 cm³. Then multiply by 4/3. Let me do that: 4/3 * 216.

Hmm, 216 divided by 3 is 72, and 72 times 4 is 288. So the volume of the sphere is 288π cm³.

Now, to find out how many beakers are needed to fill the sphere, I need to divide the volume of the sphere by the volume of one beaker.

So number of beakers = V_sphere / V_beaker = 288π / 72π.

Oh, the π cancels out, so that's 288 / 72. Let me compute that. 72 times 4 is 288, right? 72*4=288. So that would be 4.

Wait, so does that mean it takes 4 beakers to fill the spherical tank? That seems straightforward, but let me double-check my calculations to make sure I didn't make a mistake.

First, volume of the cylinder: radius 3 cm, height 8 cm. Area of the base is πr² = π*9, times height 8, so 72π. Correct.

Volume of the sphere: radius 6 cm. The formula is (4/3)πr³. So (4/3)*π*216. 216 divided by 3 is 72, times 4 is 288. So 288π. Correct.

Dividing 288π by 72π gives 4. So yes, 4 beakers.

Wait, but intuitively, a sphere and a cylinder... Is there something about the shape that might affect this? Like, maybe if you're pouring water from cylinders into a sphere, there might be some inefficiency? But the question says "how many such beakers of water will it take to fill a spherical tank," so I think it's just a matter of volume. Since volume is a measure of capacity, regardless of the shape, the number of beakers needed would just be the ratio of the volumes.

So unless there's some trick here, like maybe the beakers can't be arranged perfectly or something? But the problem doesn't mention anything like that. It just asks for the number based on volume.

So I think my answer is correct. 4 beakers.

But let me think again. Maybe I miscalculated the volumes? Let me check once more.

Cylinder: radius 3, height 8. Volume is πr²h = π*9*8 = 72π. Sphere: radius 6. Volume is (4/3)π*(6)^3. 6^3 is 216. 4/3 of 216: 216 divided by 3 is 72, times 4 is 288. So sphere is 288π. Divided by 72π is 4.

Yes, that seems right.

Alternatively, maybe the question is trying to trick me into thinking about something else? For example, if the beakers are filled to the brim and poured into the sphere, but since the sphere is a different shape, maybe there's some overlap or something? But no, volume-wise, it's just the total volume.

Alternatively, maybe the beakers are not being filled completely? But the question says "how many such beakers of water will it take to fill," so I think it's assuming that each beaker is completely filled and poured into the tank.

Therefore, I think the answer is 4.

But just to be thorough, let me think of another way. Let me compute the numerical values without π.

Volume of beaker: 72π. Volume of sphere: 288π. So ratio is 288/72 = 4.

Alternatively, if I compute the numerical values with π, but since π cancels, it's still 4.

Alternatively, maybe the question is in different units? But no, both are in cm.

Wait, the beaker is 8 cm high, radius 3 cm. Sphere radius 6 cm. So the sphere has a radius twice that of the beaker's radius, and the height of the beaker is 8 cm.

But maybe there's a relation between the dimensions? For example, if you consider the sphere's diameter is 12 cm, which is twice the radius. The beaker's height is 8 cm, which is less than the sphere's diameter.

But I don't think that affects the volume calculation.

Alternatively, maybe the question is expecting me to consider that the sphere can't be filled perfectly with cylinders? But again, unless there's some packing efficiency, which is not mentioned here, the answer should be purely based on volume.

So, I think my answer is correct. 4 beakers.

**Final Answer**
The number of beakers needed is \boxed{4}.

To determine how many cylindrical beakers are required to fill a spherical tank, we begin by calculating the volumes of both the beaker and the tank using their respective geometric formulas.

---

### Step 1: Volume of a Cylindrical Beaker

The formula for the volume of a cylinder is:

$$
V_{\text{cylinder}} = \pi r^2 h
$$

Given:
- Radius $ r = 3 \, \text{cm} $
- Height $ h = 8 \, \text{cm} $

$$
V_{\text{beaker}} = \pi (3)^2 (8) = \pi \cdot 9 \cdot 8 = 72\pi \, \text{cm}^3
$$

---

### Step 2: Volume of the Spherical Tank

The formula for the volume of a sphere is:

$$
V_{\text{sphere}} = \frac{4}{3} \pi r^3
$$

Given:
- Radius $ r = 6 \, \text{cm} $

$$
V_{\text{sphere}} = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi \cdot 216 = 288\pi \, \text{cm}^3
$$

---

### Step 3: Determine the Number of Beakers Needed

To find how many beakers are required to fill the spherical tank, divide the volume of the sphere by the volume of one beaker:

$$
\text{Number of beakers} = \frac{V_{\text{sphere}}}{V_{\text{beaker}}} = \frac{288\pi}{72\pi} = \frac{288}{72} = 4
$$

---

### Final Answer

$$
\boxed{4}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many cylindrical beakers it takes to fill a spherical tank. Let me start by recalling the formulas for the volumes of a cylinder and a sphere.

The volume of a cylinder is given by V_cylinder = πr²h, right? And the volume of a sphere is V_sphere = (4/3)πr³.

Alright, the problem says the beaker is 8 cm high and has a radius of 3 cm. So, first, I should calculate the volume of one beaker. Let me write that down:

V_beaker = π * (3 cm)² * 8 cm. Let me compute that.

First, 3 squared is 9, so that's 9 cm². Then multiply by 8 cm, so 9 * 8 = 72 cm³. Then multiply by π, so the volume of one beaker is 72π cm³.

Now, the spherical tank has a radius of 6 cm. Let me calculate its volume.

V_sphere = (4/3)π*(6 cm)³. Let me compute 6 cubed first. 6*6=36, 36*6=216. So that's 216 cm³. Then multiply by 4/3. Let me do that: 4/3 * 216.

Hmm, 216 divided by 3 is 72, and 72 times 4 is 288. So the volume of the sphere is 288π cm³.

Now, to find out how many beakers are needed to fill the sphere, I need to divide the volume of the sphere by the volume of one beaker.

So number of beakers = V_sphere / V_beaker = 288π / 72π.

Oh, the π cancels out, so that's 288 / 72. Let me compute that. 72 times 4 is 288, right? 72*4=288. So that would be 4.

Wait, so does that mean it takes 4 beakers to fill the spherical tank? That seems straightforward, but let me double-check my calculations to make sure I didn't make a mistake.

First, volume of the cylinder: radius 3 cm, height 8 cm. Area of the base is πr² = π*9, times height 8, so 72π. Correct.

Volume of the sphere: radius 6 cm. The formula is (4/3)πr³. So (4/3)*π*216. 216 divided by 3 is 72, times 4 is 288. So 288π. Correct.

Dividing 288π by 72π gives 4. So yes, 4 beakers.

Wait, but intuitively, a sphere and a cylinder... Is there something about the shape that might affect this? Like, maybe if you're pouring water from cylinders into a sphere, there might be some inefficiency? But the question says "how many such beakers of water will it take to fill a spherical tank," so I think it's just a matter of volume. Since volume is a measure of capacity, regardless of the shape, the number of beakers needed would just be the ratio of the volumes.

So unless there's some trick here, like maybe the beakers can't be arranged perfectly or something? But the problem doesn't mention anything like that. It just asks for the number based on volume.

So I think my answer is correct. 4 beakers.

But let me think again. Maybe I miscalculated the volumes? Let me check once more.

Cylinder: radius 3, height 8. Volume is πr²h = π*9*8 = 72π. Sphere: radius 6. Volume is (4/3)π*(6)^3. 6^3 is 216. 4/3 of 216: 216 divided by 3 is 72, times 4 is 288. So sphere is 288π. Divided by 72π is 4.

Yes, that seems right.

Alternatively, maybe the question is trying to trick me into thinking about something else? For example, if the beakers are filled to the brim and poured into the sphere, but since the sphere is a different shape, maybe there's some overlap or something? But no, volume-wise, it's just the total volume.

Alternatively, maybe the beakers are not being filled completely? But the question says "how many such beakers of water will it take to fill," so I think it's assuming that each beaker is completely filled and poured into the tank.

Therefore, I think the answer is 4.

But just to be thorough, let me think of another way. Let me compute the numerical values without π.

Volume of beaker: 72π. Volume of sphere: 288π. So ratio is 288/72 = 4.

Alternatively, if I compute the numerical values with π, but since π cancels, it's still 4.

Alternatively, maybe the question is in different units? But no, both are in cm.

Wait, the beaker is 8 cm high, radius 3 cm. Sphere radius 6 cm. So the sphere has a radius twice that of the beaker's radius, and the height of the beaker is 8 cm.

But maybe there's a relation between the dimensions? For example, if you consider the sphere's diameter is 12 cm, which is twice the radius. The beaker's height is 8 cm, which is less than the sphere's diameter.

But I don't think that affects the volume calculation.

Alternatively, maybe the question is expecting me to consider that the sphere can't be filled perfectly with cylinders? But again, unless there's some packing efficiency, which is not mentioned here, the answer should be purely based on volume.

So, I think my answer is correct. 4 beakers.

**Final Answer**
The number of beakers needed is \boxed{4}.

To determine how many cylindrical beakers are required to fill a spherical tank, we begin by calculating the volumes of both the beaker and the tank using their respective geometric formulas.

---

### Step 1: Volume of a Cylindrical Beaker

The formula for the volume of a cylinder is:

$$
V_{\text{cylinder}} = \pi r^2 h
$$

Given:
- Radius $ r = 3 \, \text{cm} $
- Height $ h = 8 \, \text{cm} $

$$
V_{\text{beaker}} = \pi (3)^2 (8) = \pi \cdot 9 \cdot 8 = 72\pi \, \text{cm}^3
$$

---

### Step 2: Volume of the Spherical Tank

The formula for the volume of a sphere is:

$$
V_{\text{sphere}} = \frac{4}{3} \pi r^3
$$

Given:
- Radius $ r = 6 \, \text{cm} $

$$
V_{\text{sphere}} = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi \cdot 216 = 288\pi \, \text{cm}^3
$$

---

### Step 3: Determine the Number of Beakers Needed

To find how many beakers are required to fill the spherical tank, divide the volume of the sphere by the volume of one beaker:

$$
\text{Number of beakers} = \frac{V_{\text{sphere}}}{V_{\text{beaker}}} = \frac{288\pi}{72\pi} = \frac{288}{72} = 4
$$

---

### Final Answer

$$
\boxed{4}
$$

Solution: We first have to remember our formulas for the volumes of 3 dimensional objects. The volume of a cylinder with radius $r$ and height $h$ is $r^2h\pi$ and the volume of a sphere with radius $r$ is $\frac{4}{3} r^3 \pi$. Since the cylindrical beaker has a height of 8 centimeters and a radius of 3 centimeters, that means that its volume is $3^2\cdot8\cdot\pi=72\pi$ cubic centimeters. Since the sphere has a radius of 6 centimeters, its volume is $\frac{4}{3}\cdot6^3\pi = 288\pi$ cubic centimeters. The number of beakers of what it will take to fill the spherical tank is just the ratio of the volume of the tank to the volume of the cylinder, which is given by $\dfrac{288\pi}{72\pi}=\boxed{4}$.


Explanation:
Okay, so I need to evaluate this expression: sin(arcsin 0.4 + arcsin 0.5) multiplied by sin(arcsin 0.5 - arcsin 0.4). Hmm, that seems a bit complicated at first glance, but maybe I can break it down using some trigonometric identities. Let me recall... there are formulas for sine of sum and difference of angles. Let me write them down.

The sine of a sum formula is sin(A + B) = sin A cos B + cos A sin B, and the sine of a difference is sin(A - B) = sin A cos B - cos A sin B. So maybe I can apply these formulas here. Let me denote arcsin 0.4 as α and arcsin 0.5 as β. Then the expression becomes sin(α + β) * sin(β - α). Wait, that's interesting. So the original expression is sin(α + β) multiplied by sin(β - α).

Let me compute each part step by step. First, let me find α and β. But actually, maybe I don't need to find their exact values. Since α = arcsin 0.4, that means sin α = 0.4, and similarly sin β = 0.5. Then, I can find cos α and cos β using the Pythagorean identity.

For α: cos α = sqrt(1 - sin² α) = sqrt(1 - 0.16) = sqrt(0.84). Let me compute that. sqrt(0.84)... Well, sqrt(0.81) is 0.9, sqrt(0.84) is a bit more. Maybe I can leave it as sqrt(0.84) for now or rationalize it? Let me see. 0.84 is 21/25? Wait, 0.84 is 84/100, which simplifies to 21/25. Wait, 21/25 is 0.84. So sqrt(21/25) is sqrt(21)/5. So cos α = sqrt(21)/5. Similarly for cos β: sin β = 0.5, so cos β = sqrt(1 - 0.25) = sqrt(0.75) = sqrt(3)/2.

Okay, so cos α is sqrt(21)/5 and cos β is sqrt(3)/2. Now, let me compute sin(α + β) and sin(β - α) using the formulas.

Starting with sin(α + β): that's sin α cos β + cos α sin β. Plugging in the values: sin α is 0.4, cos β is sqrt(3)/2, cos α is sqrt(21)/5, and sin β is 0.5. So:

sin(α + β) = 0.4*(sqrt(3)/2) + (sqrt(21)/5)*0.5. Let me compute each term.

First term: 0.4*(sqrt(3)/2) = (0.4/2)*sqrt(3) = 0.2*sqrt(3). Second term: (sqrt(21)/5)*0.5 = (sqrt(21)/5)*(1/2) = sqrt(21)/10. So altogether, sin(α + β) = 0.2*sqrt(3) + sqrt(21)/10.

Hmm, maybe I can write 0.2 as 1/5, so 1/5*sqrt(3) + sqrt(21)/10. To combine these terms, let me get a common denominator. The common denominator would be 10. So 1/5 is 2/10, so 2*sqrt(3)/10 + sqrt(21)/10 = [2 sqrt(3) + sqrt(21)] / 10.

Okay, so sin(α + β) is [2 sqrt(3) + sqrt(21)] / 10. Now, moving on to sin(β - α). Using the formula sin(β - α) = sin β cos α - cos β sin α. Let me compute that.

sin β is 0.5, cos α is sqrt(21)/5, cos β is sqrt(3)/2, sin α is 0.4. So:

sin(β - α) = 0.5*(sqrt(21)/5) - (sqrt(3)/2)*0.4. Let me compute each term.

First term: 0.5*(sqrt(21)/5) = (sqrt(21)/10). Second term: (sqrt(3)/2)*0.4 = (sqrt(3)/2)*(2/5) = sqrt(3)/5. Wait, 0.4 is 2/5, right? So 0.4 = 2/5. Therefore, the second term is sqrt(3)/2 * 2/5 = sqrt(3)/5.

Therefore, sin(β - α) = sqrt(21)/10 - sqrt(3)/5. Again, let me write them with common denominators. sqrt(3)/5 is 2 sqrt(3)/10, so:

sin(β - α) = sqrt(21)/10 - 2 sqrt(3)/10 = [sqrt(21) - 2 sqrt(3)] / 10.

So now, the original expression is sin(α + β) * sin(β - α) = [2 sqrt(3) + sqrt(21)] / 10 multiplied by [sqrt(21) - 2 sqrt(3)] / 10.

So that's the product of two fractions: numerator times numerator over 10*10 = 100. Let me compute the numerator: [2 sqrt(3) + sqrt(21)] * [sqrt(21) - 2 sqrt(3)].

This looks like a product of the form (a + b)(c - d), but maybe it's better to recognize it as (a + b)(a - b) if there's some symmetry? Wait, let me check. Let me denote the first term as A = 2 sqrt(3) and B = sqrt(21). Then the product is (A + B)(B - A) = B² - A².

Yes! Because (A + B)(B - A) = B² - A². So that's a difference of squares. Let me verify:

(A + B)(B - A) = A*B - A² + B² - A*B = B² - A². Yes, that's correct. So that simplifies things.

Therefore, the numerator is [sqrt(21)]² - [2 sqrt(3)]². Let me compute that.

[sqrt(21)]² is 21. [2 sqrt(3)]² is 4*(sqrt(3))² = 4*3 = 12. Therefore, the numerator is 21 - 12 = 9.

So the entire expression is 9 / 100. Therefore, the value of the original expression is 9/100, which is 0.09.

Wait, let me check my steps again to make sure I didn't make a mistake.

First, I set α = arcsin 0.4, β = arcsin 0.5. Then, sin(α + β) and sin(β - α) were calculated using the sine addition and subtraction formulas. Then, when multiplying them, I noticed that the product is of the form (A + B)(B - A) which is B² - A².

Let me check the calculation of A and B again.

Original terms:

sin(α + β) = [2 sqrt(3) + sqrt(21)] / 10.

sin(β - α) = [sqrt(21) - 2 sqrt(3)] / 10.

So when multiplying, the numerator is (2 sqrt(3) + sqrt(21))(sqrt(21) - 2 sqrt(3)).

Yes, which is (sqrt(21) + 2 sqrt(3))(sqrt(21) - 2 sqrt(3)) = (sqrt(21))² - (2 sqrt(3))² = 21 - 12 = 9.

Therefore, the numerator is 9, denominator is 10*10=100. So 9/100.

Therefore, the answer is 9/100.

But let me verify this with another approach to make sure. Maybe using another identity or calculating numerically.

Alternatively, perhaps using the identity for sin(A + B) * sin(B - A). Let me think.

Note that sin(A + B) * sin(B - A) can be written as sin(B + A) * sin(B - A). Which is similar to sin(x + y) sin(x - y) where x = B and y = A.

There is an identity for sin(x + y) sin(x - y) = sin²x - sin²y. Wait, let me recall.

Yes, using the identity: sin(x + y) sin(x - y) = sin²x - sin²y. Let me verify that.

Using the formula: sin(x + y) sin(x - y) = [sin x cos y + cos x sin y][sin x cos y - cos x sin y] = (sin x cos y)^2 - (cos x sin y)^2 = sin²x cos²y - cos²x sin²y.

Alternatively, factor it as sin²x (1 - sin²y) - (1 - sin²x) sin²y = sin²x - sin²x sin²y - sin²y + sin²x sin²y = sin²x - sin²y.

Yes! So indeed, sin(x + y) sin(x - y) = sin²x - sin²y.

Therefore, in our case, if we let x = β and y = α, then sin(β + α) sin(β - α) = sin²β - sin²α.

Wait, that's a much simpler way! So instead of computing each sine term and multiplying, we can use this identity.

So according to this identity, the original expression is sin²β - sin²α.

Given that sin β = 0.5 and sin α = 0.4, so sin²β = (0.5)^2 = 0.25, and sin²α = (0.4)^2 = 0.16. Therefore, sin²β - sin²α = 0.25 - 0.16 = 0.09, which is 9/100.

So that's the same result as before! Therefore, this confirms that the answer is indeed 9/100.

Therefore, regardless of the method, the answer is 9/100.

But just to be thorough, let me check with actual numerical values. Let me compute arcsin(0.4) and arcsin(0.5) numerically, then compute the sines of the sums and differences, multiply them, and see if it's 0.09.

First, arcsin(0.5) is π/6, which is 30 degrees. So β = π/6.

arcsin(0.4)... Let me calculate that. Let me recall that arcsin(0.4) is approximately... Well, sin(23.57 degrees) is approximately 0.4. Let me check with calculator.

But since I don't have a calculator here, but maybe I can compute it numerically. Alternatively, use approximate values.

But maybe I can use exact expressions. Wait, but if I use the identity, we already have the answer. However, for verification, let me try.

Alternatively, compute sin(arcsin 0.4 + arcsin 0.5) * sin(arcsin 0.5 - arcsin 0.4).

Let me denote α = arcsin 0.4, β = arcsin 0.5.

So, sin(α + β) * sin(β - α).

Alternatively, using the identity sin(α + β) * sin(β - α) = sin²β - sin²α.

Which is 0.25 - 0.16 = 0.09.

Alternatively, if I compute numerically:

First, compute α = arcsin(0.4). Let me approximate α.

We know that sin(π/6) = 0.5, sin(π/12) ≈ 0.2588, sin(π/6) = 0.5. So 0.4 is between π/12 and π/6. Let me use a calculator-like approach.

Let me use the Taylor series or some approximation. Alternatively, remember that sin(0.4115 radians) ≈ 0.4. Wait, but maybe I can use calculator values.

But since I don't have a calculator, maybe I can use the previous exact expressions.

But since we already have two methods giving 9/100, which is 0.09, and the identity is valid, I think that's solid.

Alternatively, let me check with approximate decimal values.

Let me compute sin(arcsin 0.4 + arcsin 0.5).

First, arcsin 0.5 is π/6 ≈ 0.5236 radians.

arcsin 0.4 is approximately... Let me recall that sin(0.4115) ≈ 0.4. Let me check with a calculator (pretending I have one). Let me use the approximation.

Alternatively, use the value of α where sin α = 0.4. Let me use the formula for sin(α + β).

But maybe I can compute sin(α + β) numerically.

But since I already have the exact value as [2 sqrt(3) + sqrt(21)] / 10. Let me compute that numerically.

First, sqrt(3) ≈ 1.732, sqrt(21) ≈ 4.5837.

So 2 sqrt(3) ≈ 3.464, sqrt(21) ≈ 4.5837.

Adding them: 3.464 + 4.5837 ≈ 8.0477. Divided by 10: ≈ 0.80477.

Then sin(α + β) ≈ 0.80477.

Then sin(β - α) was [sqrt(21) - 2 sqrt(3)] / 10.

sqrt(21) ≈ 4.5837, 2 sqrt(3) ≈ 3.464. So 4.5837 - 3.464 ≈ 1.1197. Divided by 10: ≈ 0.11197.

Then multiplying 0.80477 * 0.11197 ≈ Let me compute that.

First, 0.8 * 0.11197 ≈ 0.08958. Then 0.00477 * 0.11197 ≈ approx 0.000535. Adding together ≈ 0.08958 + 0.000535 ≈ 0.0901. Which is approximately 0.09, which is 9/100.

Therefore, this numerical approximation also gives about 0.09, which matches our exact calculation.

Therefore, I can be confident that the answer is 9/100.

Alternatively, another way to think about this problem is to recognize that the product sin(A + B) * sin(B - A) is equal to sin²B - sin²A, which is a trigonometric identity. Therefore, directly applying this identity simplifies the problem significantly.

So, in summary, by using the identity sin(x + y) sin(x - y) = sin²x - sin²y, we can directly compute the value as (sin²(arcsin 0.5) - sin²(arcsin 0.4)) = (0.5² - 0.4²) = 0.25 - 0.16 = 0.09 = 9/100.

Therefore, the answer is 9/100.

**Final Answer**
\boxed{\dfrac{9}{100}}

To evaluate the expression:

$$
\sin (\arcsin 0.4 + \arcsin 0.5) \cdot \sin (\arcsin 0.5 - \arcsin 0.4),
$$

we can simplify the problem using a key trigonometric identity:

$$
\sin(A + B) \cdot \sin(B - A) = \sin^2 B - \sin^2 A.
$$

---

### Step 1: Apply the Identity

Let:
- $ A = \arcsin 0.4 $
- $ B = \arcsin 0.5 $

Then the expression becomes:

$$
\sin(\arcsin 0.4 + \arcsin 0.5) \cdot \sin(\arcsin 0.5 - \arcsin 0.4) = \sin^2(\arcsin 0.5) - \sin^2(\arcsin 0.4)
$$

---

### Step 2: Use the Identity $ \sin(\arcsin x) = x $

Since $ \sin(\arcsin x) = x $, we have:

$$
\sin^2(\arcsin 0.5) = (0.5)^2 = 0.25, \quad \sin^2(\arcsin 0.4) = (0.4)^2 = 0.16
$$

So the expression becomes:

$$
0.25 - 0.16 = 0.09
$$

---

### Step 3: Final Answer

$$
\boxed{\dfrac{9}{100}}
$$

[Assistant]
Explanation:
Okay, so I need to evaluate this expression: sin(arcsin 0.4 + arcsin 0.5) multiplied by sin(arcsin 0.5 - arcsin 0.4). Hmm, that seems a bit complicated at first glance, but maybe I can break it down using some trigonometric identities. Let me recall... there are formulas for sine of sum and difference of angles. Let me write them down.

The sine of a sum formula is sin(A + B) = sin A cos B + cos A sin B, and the sine of a difference is sin(A - B) = sin A cos B - cos A sin B. So maybe I can apply these formulas here. Let me denote arcsin 0.4 as α and arcsin 0.5 as β. Then the expression becomes sin(α + β) * sin(β - α). Wait, that's interesting. So the original expression is sin(α + β) multiplied by sin(β - α).

Let me compute each part step by step. First, let me find α and β. But actually, maybe I don't need to find their exact values. Since α = arcsin 0.4, that means sin α = 0.4, and similarly sin β = 0.5. Then, I can find cos α and cos β using the Pythagorean identity.

For α: cos α = sqrt(1 - sin² α) = sqrt(1 - 0.16) = sqrt(0.84). Let me compute that. sqrt(0.84)... Well, sqrt(0.81) is 0.9, sqrt(0.84) is a bit more. Maybe I can leave it as sqrt(0.84) for now or rationalize it? Let me see. 0.84 is 21/25? Wait, 0.84 is 84/100, which simplifies to 21/25. Wait, 21/25 is 0.84. So sqrt(21/25) is sqrt(21)/5. So cos α = sqrt(21)/5. Similarly for cos β: sin β = 0.5, so cos β = sqrt(1 - 0.25) = sqrt(0.75) = sqrt(3)/2.

Okay, so cos α is sqrt(21)/5 and cos β is sqrt(3)/2. Now, let me compute sin(α + β) and sin(β - α) using the formulas.

Starting with sin(α + β): that's sin α cos β + cos α sin β. Plugging in the values: sin α is 0.4, cos β is sqrt(3)/2, cos α is sqrt(21)/5, and sin β is 0.5. So:

sin(α + β) = 0.4*(sqrt(3)/2) + (sqrt(21)/5)*0.5. Let me compute each term.

First term: 0.4*(sqrt(3)/2) = (0.4/2)*sqrt(3) = 0.2*sqrt(3). Second term: (sqrt(21)/5)*0.5 = (sqrt(21)/5)*(1/2) = sqrt(21)/10. So altogether, sin(α + β) = 0.2*sqrt(3) + sqrt(21)/10.

Hmm, maybe I can write 0.2 as 1/5, so 1/5*sqrt(3) + sqrt(21)/10. To combine these terms, let me get a common denominator. The common denominator would be 10. So 1/5 is 2/10, so 2*sqrt(3)/10 + sqrt(21)/10 = [2 sqrt(3) + sqrt(21)] / 10.

Okay, so sin(α + β) is [2 sqrt(3) + sqrt(21)] / 10. Now, moving on to sin(β - α). Using the formula sin(β - α) = sin β cos α - cos β sin α. Let me compute that.

sin β is 0.5, cos α is sqrt(21)/5, cos β is sqrt(3)/2, sin α is 0.4. So:

sin(β - α) = 0.5*(sqrt(21)/5) - (sqrt(3)/2)*0.4. Let me compute each term.

First term: 0.5*(sqrt(21)/5) = (sqrt(21)/10). Second term: (sqrt(3)/2)*0.4 = (sqrt(3)/2)*(2/5) = sqrt(3)/5. Wait, 0.4 is 2/5, right? So 0.4 = 2/5. Therefore, the second term is sqrt(3)/2 * 2/5 = sqrt(3)/5.

Therefore, sin(β - α) = sqrt(21)/10 - sqrt(3)/5. Again, let me write them with common denominators. sqrt(3)/5 is 2 sqrt(3)/10, so:

sin(β - α) = sqrt(21)/10 - 2 sqrt(3)/10 = [sqrt(21) - 2 sqrt(3)] / 10.

So now, the original expression is sin(α + β) * sin(β - α) = [2 sqrt(3) + sqrt(21)] / 10 multiplied by [sqrt(21) - 2 sqrt(3)] / 10.

So that's the product of two fractions: numerator times numerator over 10*10 = 100. Let me compute the numerator: [2 sqrt(3) + sqrt(21)] * [sqrt(21) - 2 sqrt(3)].

This looks like a product of the form (a + b)(c - d), but maybe it's better to recognize it as (a + b)(a - b) if there's some symmetry? Wait, let me check. Let me denote the first term as A = 2 sqrt(3) and B = sqrt(21). Then the product is (A + B)(B - A) = B² - A².

Yes! Because (A + B)(B - A) = B² - A². So that's a difference of squares. Let me verify:

(A + B)(B - A) = A*B - A² + B² - A*B = B² - A². Yes, that's correct. So that simplifies things.

Therefore, the numerator is [sqrt(21)]² - [2 sqrt(3)]². Let me compute that.

[sqrt(21)]² is 21. [2 sqrt(3)]² is 4*(sqrt(3))² = 4*3 = 12. Therefore, the numerator is 21 - 12 = 9.

So the entire expression is 9 / 100. Therefore, the value of the original expression is 9/100, which is 0.09.

Wait, let me check my steps again to make sure I didn't make a mistake.

First, I set α = arcsin 0.4, β = arcsin 0.5. Then, sin(α + β) and sin(β - α) were calculated using the sine addition and subtraction formulas. Then, when multiplying them, I noticed that the product is of the form (A + B)(B - A) which is B² - A².

Let me check the calculation of A and B again.

Original terms:

sin(α + β) = [2 sqrt(3) + sqrt(21)] / 10.

sin(β - α) = [sqrt(21) - 2 sqrt(3)] / 10.

So when multiplying, the numerator is (2 sqrt(3) + sqrt(21))(sqrt(21) - 2 sqrt(3)).

Yes, which is (sqrt(21) + 2 sqrt(3))(sqrt(21) - 2 sqrt(3)) = (sqrt(21))² - (2 sqrt(3))² = 21 - 12 = 9.

Therefore, the numerator is 9, denominator is 10*10=100. So 9/100.

Therefore, the answer is 9/100.

But let me verify this with another approach to make sure. Maybe using another identity or calculating numerically.

Alternatively, perhaps using the identity for sin(A + B) * sin(B - A). Let me think.

Note that sin(A + B) * sin(B - A) can be written as sin(B + A) * sin(B - A). Which is similar to sin(x + y) sin(x - y) where x = B and y = A.

There is an identity for sin(x + y) sin(x - y) = sin²x - sin²y. Wait, let me recall.

Yes, using the identity: sin(x + y) sin(x - y) = sin²x - sin²y. Let me verify that.

Using the formula: sin(x + y) sin(x - y) = [sin x cos y + cos x sin y][sin x cos y - cos x sin y] = (sin x cos y)^2 - (cos x sin y)^2 = sin²x cos²y - cos²x sin²y.

Alternatively, factor it as sin²x (1 - sin²y) - (1 - sin²x) sin²y = sin²x - sin²x sin²y - sin²y + sin²x sin²y = sin²x - sin²y.

Yes! So indeed, sin(x + y) sin(x - y) = sin²x - sin²y.

Therefore, in our case, if we let x = β and y = α, then sin(β + α) sin(β - α) = sin²β - sin²α.

Wait, that's a much simpler way! So instead of computing each sine term and multiplying, we can use this identity.

So according to this identity, the original expression is sin²β - sin²α.

Given that sin β = 0.5 and sin α = 0.4, so sin²β = (0.5)^2 = 0.25, and sin²α = (0.4)^2 = 0.16. Therefore, sin²β - sin²α = 0.25 - 0.16 = 0.09, which is 9/100.

So that's the same result as before! Therefore, this confirms that the answer is indeed 9/100.

Therefore, regardless of the method, the answer is 9/100.

But just to be thorough, let me check with actual numerical values. Let me compute arcsin(0.4) and arcsin(0.5) numerically, then compute the sines of the sums and differences, multiply them, and see if it's 0.09.

First, arcsin(0.5) is π/6, which is 30 degrees. So β = π/6.

arcsin(0.4)... Let me calculate that. Let me recall that arcsin(0.4) is approximately... Well, sin(23.57 degrees) is approximately 0.4. Let me check with calculator.

But since I don't have a calculator here, but maybe I can compute it numerically. Alternatively, use approximate values.

But maybe I can use exact expressions. Wait, but if I use the identity, we already have the answer. However, for verification, let me try.

Alternatively, compute sin(arcsin 0.4 + arcsin 0.5) * sin(arcsin 0.5 - arcsin 0.4).

Let me denote α = arcsin 0.4, β = arcsin 0.5.

So, sin(α + β) * sin(β - α).

Alternatively, using the identity sin(α + β) * sin(β - α) = sin²β - sin²α.

Which is 0.25 - 0.16 = 0.09.

Alternatively, if I compute numerically:

First, compute α = arcsin(0.4). Let me approximate α.

We know that sin(π/6) = 0.5, sin(π/12) ≈ 0.2588, sin(π/6) = 0.5. So 0.4 is between π/12 and π/6. Let me use a calculator-like approach.

Let me use the Taylor series or some approximation. Alternatively, remember that sin(0.4115 radians) ≈ 0.4. Wait, but maybe I can use calculator values.

But since I don't have a calculator, maybe I can use the previous exact expressions.

But since we already have two methods giving 9/100, which is 0.09, and the identity is valid, I think that's solid.

Alternatively, let me check with approximate decimal values.

Let me compute sin(arcsin 0.4 + arcsin 0.5).

First, arcsin 0.5 is π/6 ≈ 0.5236 radians.

arcsin 0.4 is approximately... Let me recall that sin(0.4115) ≈ 0.4. Let me check with a calculator (pretending I have one). Let me use the approximation.

Alternatively, use the value of α where sin α = 0.4. Let me use the formula for sin(α + β).

But maybe I can compute sin(α + β) numerically.

But since I already have the exact value as [2 sqrt(3) + sqrt(21)] / 10. Let me compute that numerically.

First, sqrt(3) ≈ 1.732, sqrt(21) ≈ 4.5837.

So 2 sqrt(3) ≈ 3.464, sqrt(21) ≈ 4.5837.

Adding them: 3.464 + 4.5837 ≈ 8.0477. Divided by 10: ≈ 0.80477.

Then sin(α + β) ≈ 0.80477.

Then sin(β - α) was [sqrt(21) - 2 sqrt(3)] / 10.

sqrt(21) ≈ 4.5837, 2 sqrt(3) ≈ 3.464. So 4.5837 - 3.464 ≈ 1.1197. Divided by 10: ≈ 0.11197.

Then multiplying 0.80477 * 0.11197 ≈ Let me compute that.

First, 0.8 * 0.11197 ≈ 0.08958. Then 0.00477 * 0.11197 ≈ approx 0.000535. Adding together ≈ 0.08958 + 0.000535 ≈ 0.0901. Which is approximately 0.09, which is 9/100.

Therefore, this numerical approximation also gives about 0.09, which matches our exact calculation.

Therefore, I can be confident that the answer is 9/100.

Alternatively, another way to think about this problem is to recognize that the product sin(A + B) * sin(B - A) is equal to sin²B - sin²A, which is a trigonometric identity. Therefore, directly applying this identity simplifies the problem significantly.

So, in summary, by using the identity sin(x + y) sin(x - y) = sin²x - sin²y, we can directly compute the value as (sin²(arcsin 0.5) - sin²(arcsin 0.4)) = (0.5² - 0.4²) = 0.25 - 0.16 = 0.09 = 9/100.

Therefore, the answer is 9/100.

**Final Answer**
\boxed{\dfrac{9}{100}}

To evaluate the expression:

$$
\sin (\arcsin 0.4 + \arcsin 0.5) \cdot \sin (\arcsin 0.5 - \arcsin 0.4),
$$

we can simplify the problem using a key trigonometric identity:

$$
\sin(A + B) \cdot \sin(B - A) = \sin^2 B - \sin^2 A.
$$

---

### Step 1: Apply the Identity

Let:
- $ A = \arcsin 0.4 $
- $ B = \arcsin 0.5 $

Then the expression becomes:

$$
\sin(\arcsin 0.4 + \arcsin 0.5) \cdot \sin(\arcsin 0.5 - \arcsin 0.4) = \sin^2(\arcsin 0.5) - \sin^2(\arcsin 0.4)
$$

---

### Step 2: Use the Identity $ \sin(\arcsin x) = x $

Since $ \sin(\arcsin x) = x $, we have:

$$
\sin^2(\arcsin 0.5) = (0.5)^2 = 0.25, \quad \sin^2(\arcsin 0.4) = (0.4)^2 = 0.16
$$

So the expression becomes:

$$
0.25 - 0.16 = 0.09
$$

---

### Step 3: Final Answer

$$
\boxed{\dfrac{9}{100}}
$$

Solution:
From the angle addition and subtraction formulas,
\begin{align*}
\sin (x + y) &= \sin x \cos y + \cos x \sin y, \\
\sin (x - y) &= \sin x \cos y - \cos x \sin y,
\end{align*}so
\begin{align*}
\sin (x + y) \sin (x - y) &= (\sin x \cos y + \cos x \sin y)(\sin x \cos y - \cos x \sin y) \\
&= \sin^2 x \cos^2 y + \sin x \cos x \sin y \cos y - \sin x \cos x \sin y \cos y - \cos^2 x \sin^2 y \\
&= \sin^2 x (1 - \sin^2 y) - (1 - \sin^2 x) \sin^2 y \\
&= \sin^2 x - \sin^2 x \sin^2 y - \sin^2 y + \sin^2 x \sin^2 y \\
&= \sin^2 x - \sin^2 y.
\end{align*}Taking $x = \arcsin 0.5$ and $y = \arcsin 0.4,$ we get
\begin{align*}
\sin (\arcsin 0.5 + \arcsin 0.4) \cdot \sin (\arcsin 0.5 - \arcsin 0.4) &= \sin^2 (\arcsin 0.5) - \sin^2 (\arcsin 0.4) \\
&= 0.5^2 - 0.4^2 \\
&= 0.09 = \boxed{\frac{9}{100}}.
\end{align*}


Explanation:
Okay, so I need to figure out how many different license plates can be made with the given conditions. Let me start by understanding the problem again. The license plate has 2 different letters followed by 2 different one-digit numbers. The letters are from A through Z, which I know there are 26 letters in the English alphabet. The numbers are from 0 to 9, so that's 10 different digits.

First, let me break down the problem into parts. The license plate has two sections: letters and numbers. The letters part has two different letters, and the numbers part has two different digits. So I need to calculate the number of possibilities for the letters and then the number for the digits, and then multiply them together because for each combination of letters, there are multiple combinations of numbers, and vice versa.

Starting with the letters. The first thing I need to consider is that the two letters must be different. So, for the first letter, there are 26 possible choices (A-Z). Once I choose the first letter, the second letter has to be different from the first one. Therefore, there are 25 remaining choices for the second letter.

Wait, but hold on—is the order important here? Like, is "AB" considered different from "BA"? I think in license plates, the order does matter. For example, if the first letter is A and the second is B, that's a different plate than first B and then A. So yes, order matters here. Therefore, the number of possible combinations for the letters is 26 * 25.

Let me verify that. If I have 26 choices for the first letter and 25 for the second, then multiplying them gives the total number of ordered pairs where the two letters are different. That seems right. So letters part: 26 * 25.

Now moving on to the numbers part. The problem says two different one-digit numbers. So similar logic applies here. The first digit can be any of the 10 digits (0-9). Then the second digit has to be different from the first one. Therefore, there are 9 choices left for the second digit.

Again, order matters here as well. For example, "12" is different from "21". Therefore, the number of possible combinations for the numbers is 10 * 9.

So then, the total number of license plates would be the number of letter combinations multiplied by the number of number combinations. That is (26 * 25) * (10 * 9). Let me compute that.

First, let me calculate 26 * 25. 26 times 25. Well, 25*26 is the same as 25*(20 + 6) = 25*20 + 25*6 = 500 + 150 = 650. So letters part is 650.

Then numbers part: 10 * 9 = 90.

Therefore, total license plates would be 650 * 90. Let me compute that.

Breaking it down: 650 * 90. Well, 650 * 9 * 10. 650 * 9. Let me compute 650 * 9.

First, 600 * 9 = 5400, and 50 * 9 = 450. So 5400 + 450 = 5850. Then multiply by 10: 5850 * 10 = 58,500.

So total number of license plates is 58,500.

Wait, let me check if I did that correctly. Alternatively, 650 * 90. Since 650 * 10 = 6500, so 650 * 90 = 650 * 10 * 9 = 6500 * 9. Let me compute 6500 * 9.

Well, 6000 * 9 = 54,000 and 500 * 9 = 4,500. Adding them together: 54,000 + 4,500 = 58,500. Yep, same result.

So that seems to check out.

But let me think again if there's another way to approach this problem, just to verify.

Alternatively, for the letters, since they are two different letters, the number of permutations of 26 letters taken 2 at a time is P(26, 2) = 26 * 25 = 650, which matches what I had before. Similarly, for the numbers, it's permutations of 10 digits taken 2 at a time, which is P(10, 2) = 10 * 9 = 90. Then total number is 650 * 90 = 58,500.

So that seems consistent.

Alternatively, if I thought about combinations instead of permutations, but since the order matters in the license plate (i.e., AB is different from BA), we can't use combinations. Combinations would be if the order didn't matter, but here it does. So permutations are the right approach.

Another check: if the letters were allowed to repeat, then it would be 26 * 26 for letters, and 10 * 10 for numbers, giving 676 * 100 = 67,600. But since they have to be different, we have fewer possibilities. So 58,500 is less than 67,600, which makes sense.

Similarly, if the numbers were allowed to repeat, it would be 10 * 10 = 100, so 650 * 100 = 65,000. But since numbers can't repeat, it's 90 instead of 100, so 650 * 90 = 58,500. That also makes sense.

So all these cross-checks seem to confirm that 58,500 is the correct number.

Wait, but let me think again if there's any other constraints or if I misread the problem. The problem says "2 different letters (A through Z) followed by 2 different one digit (0-9) numbers". So, letters are different, numbers are different. No other constraints. So, for example, the letters can be any two different letters, and numbers can be any two different digits.

So, for example, a license plate like "AB12" is allowed, as well as "BA21", etc. So order matters.

Another thing to check: are the numbers allowed to start with zero? For example, is "01" considered a valid number part? The problem says "one digit numbers", but since they are two different digits, but digits can be 0-9. So yes, 0 is allowed. So "01" is okay. So no problem there.

So, the calculation seems okay.

Alternatively, if someone thought that the numbers have to be two-digit numbers, but the problem says "2 different one digit numbers", so each is a single digit, just two different ones. So, for example, the numbers part is two digits, each from 0-9, different from each other. So, for example, 12, 21, 01, etc., all allowed.

Therefore, the calculation is correct.

So, I think the answer is 58,500.

But just to make sure, let me do the multiplication again.

26 * 25 = 650.

10 * 9 = 90.

650 * 90.

Well, 650 * 90 = 650 * 9 * 10.

650 * 9:

Let me compute 650 * 9:

First, 600 * 9 = 5,400.

Then 50 * 9 = 450.

Adding them together: 5,400 + 450 = 5,850.

Then multiply by 10: 5,850 * 10 = 58,500.

Yes, same result.

Alternatively, 650 * 90:

Multiply 650 by 90:

Breakdown 90 into 9 * 10, as before.

Alternatively, think of 650 * 90 as 65 * 10 * 9 * 10 = 65 * 9 * 100.

65 * 9: 60*9 + 5*9 = 540 + 45 = 585.

Then 585 * 100 = 58,500.

Same answer.

Therefore, I think I can be confident that the total number of license plates is 58,500.

I don't see any mistakes in my reasoning. The key was recognizing that both letters and numbers are permutations (since order matters) and that repetition is not allowed. So, permutations without repetition for both parts, then multiplying them together.

Another way to think about permutations:

For the letters, the number of permutations is 26P2 = 26! / (26 - 2)! = 26! / 24! = 26 * 25 * 24! / 24! = 26 * 25 = 650.

Similarly, for numbers: 10P2 = 10! / (10 - 2)! = 10! / 8! = 10 * 9 * 8! / 8! = 10 * 9 = 90.

Therefore, total number is 650 * 90 = 58,500.

Yep, that's the same as before.

So, I think that's solid.

**Final Answer**
The number of possible license plates is \boxed{58500}.

To determine how many different license plates can be formed under the given conditions, we analyze the structure of the license plate and compute the number of valid combinations for each part.

---

### **1. Understanding the Structure**

Each license plate consists of:

- **Two different letters** from A to Z (26 letters total)
- **Two different digits** from 0 to 9 (10 digits total)

The order of both the letters and the digits matters, and **no repetition is allowed** within each group.

---

### **2. Calculating the Number of Letter Combinations**

We are choosing **2 different letters** from 26, and the order matters. This is a **permutation** of 26 letters taken 2 at a time:

$$
P(26, 2) = 26 \times 25 = 650
$$

So, there are **650** valid combinations for the letter part.

---

### **3. Calculating the Number of Digit Combinations**

Similarly, we are choosing **2 different digits** from 10, and the order matters. This is a **permutation** of 10 digits taken 2 at a time:

$$
P(10, 2) = 10 \times 9 = 90
$$

So, there are **90** valid combinations for the digit part.

---

### **4. Total Number of License Plates**

Since the letter and digit parts are independent, the total number of license plates is the product of the two:

$$
\text{Total} = 650 \times 90 = 58,500
$$

---

### **Final Answer**

$$
\boxed{58500}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many different license plates can be made with the given conditions. Let me start by understanding the problem again. The license plate has 2 different letters followed by 2 different one-digit numbers. The letters are from A through Z, which I know there are 26 letters in the English alphabet. The numbers are from 0 to 9, so that's 10 different digits.

First, let me break down the problem into parts. The license plate has two sections: letters and numbers. The letters part has two different letters, and the numbers part has two different digits. So I need to calculate the number of possibilities for the letters and then the number for the digits, and then multiply them together because for each combination of letters, there are multiple combinations of numbers, and vice versa.

Starting with the letters. The first thing I need to consider is that the two letters must be different. So, for the first letter, there are 26 possible choices (A-Z). Once I choose the first letter, the second letter has to be different from the first one. Therefore, there are 25 remaining choices for the second letter.

Wait, but hold on—is the order important here? Like, is "AB" considered different from "BA"? I think in license plates, the order does matter. For example, if the first letter is A and the second is B, that's a different plate than first B and then A. So yes, order matters here. Therefore, the number of possible combinations for the letters is 26 * 25.

Let me verify that. If I have 26 choices for the first letter and 25 for the second, then multiplying them gives the total number of ordered pairs where the two letters are different. That seems right. So letters part: 26 * 25.

Now moving on to the numbers part. The problem says two different one-digit numbers. So similar logic applies here. The first digit can be any of the 10 digits (0-9). Then the second digit has to be different from the first one. Therefore, there are 9 choices left for the second digit.

Again, order matters here as well. For example, "12" is different from "21". Therefore, the number of possible combinations for the numbers is 10 * 9.

So then, the total number of license plates would be the number of letter combinations multiplied by the number of number combinations. That is (26 * 25) * (10 * 9). Let me compute that.

First, let me calculate 26 * 25. 26 times 25. Well, 25*26 is the same as 25*(20 + 6) = 25*20 + 25*6 = 500 + 150 = 650. So letters part is 650.

Then numbers part: 10 * 9 = 90.

Therefore, total license plates would be 650 * 90. Let me compute that.

Breaking it down: 650 * 90. Well, 650 * 9 * 10. 650 * 9. Let me compute 650 * 9.

First, 600 * 9 = 5400, and 50 * 9 = 450. So 5400 + 450 = 5850. Then multiply by 10: 5850 * 10 = 58,500.

So total number of license plates is 58,500.

Wait, let me check if I did that correctly. Alternatively, 650 * 90. Since 650 * 10 = 6500, so 650 * 90 = 650 * 10 * 9 = 6500 * 9. Let me compute 6500 * 9.

Well, 6000 * 9 = 54,000 and 500 * 9 = 4,500. Adding them together: 54,000 + 4,500 = 58,500. Yep, same result.

So that seems to check out.

But let me think again if there's another way to approach this problem, just to verify.

Alternatively, for the letters, since they are two different letters, the number of permutations of 26 letters taken 2 at a time is P(26, 2) = 26 * 25 = 650, which matches what I had before. Similarly, for the numbers, it's permutations of 10 digits taken 2 at a time, which is P(10, 2) = 10 * 9 = 90. Then total number is 650 * 90 = 58,500.

So that seems consistent.

Alternatively, if I thought about combinations instead of permutations, but since the order matters in the license plate (i.e., AB is different from BA), we can't use combinations. Combinations would be if the order didn't matter, but here it does. So permutations are the right approach.

Another check: if the letters were allowed to repeat, then it would be 26 * 26 for letters, and 10 * 10 for numbers, giving 676 * 100 = 67,600. But since they have to be different, we have fewer possibilities. So 58,500 is less than 67,600, which makes sense.

Similarly, if the numbers were allowed to repeat, it would be 10 * 10 = 100, so 650 * 100 = 65,000. But since numbers can't repeat, it's 90 instead of 100, so 650 * 90 = 58,500. That also makes sense.

So all these cross-checks seem to confirm that 58,500 is the correct number.

Wait, but let me think again if there's any other constraints or if I misread the problem. The problem says "2 different letters (A through Z) followed by 2 different one digit (0-9) numbers". So, letters are different, numbers are different. No other constraints. So, for example, the letters can be any two different letters, and numbers can be any two different digits.

So, for example, a license plate like "AB12" is allowed, as well as "BA21", etc. So order matters.

Another thing to check: are the numbers allowed to start with zero? For example, is "01" considered a valid number part? The problem says "one digit numbers", but since they are two different digits, but digits can be 0-9. So yes, 0 is allowed. So "01" is okay. So no problem there.

So, the calculation seems okay.

Alternatively, if someone thought that the numbers have to be two-digit numbers, but the problem says "2 different one digit numbers", so each is a single digit, just two different ones. So, for example, the numbers part is two digits, each from 0-9, different from each other. So, for example, 12, 21, 01, etc., all allowed.

Therefore, the calculation is correct.

So, I think the answer is 58,500.

But just to make sure, let me do the multiplication again.

26 * 25 = 650.

10 * 9 = 90.

650 * 90.

Well, 650 * 90 = 650 * 9 * 10.

650 * 9:

Let me compute 650 * 9:

First, 600 * 9 = 5,400.

Then 50 * 9 = 450.

Adding them together: 5,400 + 450 = 5,850.

Then multiply by 10: 5,850 * 10 = 58,500.

Yes, same result.

Alternatively, 650 * 90:

Multiply 650 by 90:

Breakdown 90 into 9 * 10, as before.

Alternatively, think of 650 * 90 as 65 * 10 * 9 * 10 = 65 * 9 * 100.

65 * 9: 60*9 + 5*9 = 540 + 45 = 585.

Then 585 * 100 = 58,500.

Same answer.

Therefore, I think I can be confident that the total number of license plates is 58,500.

I don't see any mistakes in my reasoning. The key was recognizing that both letters and numbers are permutations (since order matters) and that repetition is not allowed. So, permutations without repetition for both parts, then multiplying them together.

Another way to think about permutations:

For the letters, the number of permutations is 26P2 = 26! / (26 - 2)! = 26! / 24! = 26 * 25 * 24! / 24! = 26 * 25 = 650.

Similarly, for numbers: 10P2 = 10! / (10 - 2)! = 10! / 8! = 10 * 9 * 8! / 8! = 10 * 9 = 90.

Therefore, total number is 650 * 90 = 58,500.

Yep, that's the same as before.

So, I think that's solid.

**Final Answer**
The number of possible license plates is \boxed{58500}.

To determine how many different license plates can be formed under the given conditions, we analyze the structure of the license plate and compute the number of valid combinations for each part.

---

### **1. Understanding the Structure**

Each license plate consists of:

- **Two different letters** from A to Z (26 letters total)
- **Two different digits** from 0 to 9 (10 digits total)

The order of both the letters and the digits matters, and **no repetition is allowed** within each group.

---

### **2. Calculating the Number of Letter Combinations**

We are choosing **2 different letters** from 26, and the order matters. This is a **permutation** of 26 letters taken 2 at a time:

$$
P(26, 2) = 26 \times 25 = 650
$$

So, there are **650** valid combinations for the letter part.

---

### **3. Calculating the Number of Digit Combinations**

Similarly, we are choosing **2 different digits** from 10, and the order matters. This is a **permutation** of 10 digits taken 2 at a time:

$$
P(10, 2) = 10 \times 9 = 90
$$

So, there are **90** valid combinations for the digit part.

---

### **4. Total Number of License Plates**

Since the letter and digit parts are independent, the total number of license plates is the product of the two:

$$
\text{Total} = 650 \times 90 = 58,500
$$

---

### **Final Answer**

$$
\boxed{58500}
$$

Solution: The first letter can be any of the 26 letters of the alphabet, while the second letter can be any of the 25 remaining letters. The first digit can be any of the 10 digits, while the second digit can be any of the 9 remaining digits. The number of license plates is $26\times 25\times 10\times 9=\boxed{58,500}$.


Explanation:
Okay, so I need to find the value of 'a' that makes the function f(n) continuous. The function is defined piecewise: it's 4n + 3 when n is less than a, and 7n - 12 when n is greater than or equal to a. Hmm, continuity means that there are no jumps or breaks in the graph, right? So, at the point where the definition of the function changes, which is at n = a, the two pieces should meet. That means the value of the function from the left (when n approaches a from below) should be equal to the value from the right (when n approaches a from above).

Let me write that down. For continuity at n = a, the limit as n approaches a from the left should equal the limit as n approaches a from the right, and both should equal f(a). Since the function is defined differently on either side of a, I need to set the two expressions equal to each other at n = a.

So, if I plug n = a into both expressions, they should be the same. Let me check that.

First, when n is less than a, the function is 4n + 3. So, approaching a from the left, the value would be 4a + 3.

On the other hand, when n is greater than or equal to a, the function is 7n - 12. So, approaching a from the right, the value would be 7a - 12.

Since the function is continuous at a, these two expressions must be equal. Therefore, I can set up the equation:

4a + 3 = 7a - 12

Now, I need to solve for a. Let me subtract 4a from both sides:

3 = 3a - 12

Then, add 12 to both sides:

15 = 3a

Divide both sides by 3:

a = 5

Wait, let me check that again. If a is 5, then let's verify if the function is continuous there.

For n approaching 5 from the left, f(n) approaches 4*5 + 3 = 20 + 3 = 23.

For n approaching 5 from the right, f(n) approaches 7*5 - 12 = 35 - 12 = 23.

And f(5) itself is 7*5 - 12 = 23. So yes, both sides agree. Therefore, the function is continuous at a = 5.

But wait, let me make sure there isn't any other condition or something I missed. The problem says "the graph of y = f(n) is continuous." So, does that mean continuous everywhere? Well, since the function is defined as two linear functions, which are continuous everywhere on their own. The only potential point of discontinuity is at n = a, so by ensuring continuity at n = a, the entire function is continuous.

Therefore, the answer should be a = 5.

But let me think again. Is there a possibility that the function could have another point where it's discontinuous? For example, if there was a different definition for other values? But according to the problem, the function is defined as 4n + 3 when n < a and 7n - 12 when n ≥ a. So, there's no other points of definition. So, the only point where the function could be discontinuous is at n = a. Therefore, solving for a such that the left-hand limit equals the right-hand limit (and f(a)) is sufficient.

Therefore, my conclusion is that a = 5.

But just to be thorough, let me consider if there's any other way this could be approached. For example, maybe the problem is in terms of integers? Wait, the problem didn't specify that n is an integer. It just says n. So, n is a real number? Or is n an integer? Hmm, the problem didn't specify. But since it's talking about the graph of y = f(n), which is typically a function of a real variable. If n were an integer, the problem might have mentioned it, like "for integer values of n" or something. So, I think n is a real number here.

Therefore, the continuity condition is as I considered before. So, solving 4a + 3 = 7a - 12 gives a = 5.

Alternatively, if n were an integer, then the function would be defined for integer n, and the continuity would be about the function being continuous at integer points. But since the problem says "the graph of y = f(n) is continuous," which usually implies continuity for all real numbers, unless specified otherwise.

But even if n were an integer, the function would be continuous at each integer point. However, since the function is defined piecewise with different expressions for n < a and n ≥ a, the only point where the function could have a discontinuity is at n = a. Therefore, regardless of whether n is real or integer, the condition for continuity at n = a is the same.

Therefore, I think my answer is correct.

Wait, but if n is an integer, then the function is defined for integers. So, for example, if a is an integer, then for n approaching a from the left (i.e., n = a - 1), the function would be 4(a - 1) + 3, and for n = a, it's 7a - 12. So, in that case, the continuity would require that 4(a - 1) + 3 = 7a - 12. Let me check that.

If n is integer, then the left-hand limit as n approaches a from the left would be f(a - 1) = 4(a - 1) + 3 = 4a - 4 + 3 = 4a - 1. The right-hand limit is f(a) = 7a - 12. So, setting them equal: 4a - 1 = 7a - 12. Then, solving: -1 + 12 = 7a - 4a => 11 = 3a => a = 11/3 ≈ 3.666... But if a is supposed to be an integer, then this would not be possible. However, the problem didn't specify that a is an integer.

But the problem says "Find a if the graph of y = f(n) is continuous." If n is an integer, then the graph would consist of discrete points, but the term "continuous" is usually applied to functions defined on real numbers. So, if the problem is in the context of real numbers, then my original answer of a = 5 is correct. If it's in the context of integers, then the problem might be different, but since the problem didn't specify, I think the safe assumption is that n is a real number.

Therefore, the answer is a = 5.

Another way to check: if a = 5, then the function is 4n + 3 for n < 5 and 7n - 12 for n ≥ 5. Let me pick a value less than 5, say n = 4. Then f(4) = 4*4 + 3 = 19. For n = 5, f(5) = 7*5 - 12 = 35 - 12 = 23. If I check the value approaching 5 from the left, say n = 4.9, f(4.9) = 4*(4.9) + 3 = 19.6 + 3 = 22.6. Approaching from the right, n = 5.1, f(5.1) = 7*(5.1) - 12 = 35.7 - 12 = 23.7. Wait, but hold on, if a = 5, then at n = 5, the function is 7n - 12, which is 23. But approaching from the left, the limit is 4*5 + 3 = 23. So, the left limit is 23, the right limit is 7*5 - 12 = 23, and f(5) is 23. So, that's continuous.

But when I checked n = 4.9, which is approaching 5 from the left, I get 22.6, which is approaching 23. Similarly, n = 5.1 gives 23.7, which is approaching 23? Wait, no. Wait, 7*5.1 is 35.7, minus 12 is 23.7. Wait, but if n approaches 5 from the right, then as n approaches 5, 7n - 12 approaches 35 - 12 = 23. So, 23.7 is when n is 5.1, which is further away. So, as n approaches 5 from the right, the value approaches 23. Similarly, approaching from the left, approaching 5, the value approaches 23. Therefore, the function is continuous at 5.

So, my initial calculation was correct. Therefore, a = 5 is the correct answer.

I think I've checked it thoroughly. So, I think the answer is 5.

**Final Answer**
The value of $ a $ that makes the graph of $ y = f(n) $ continuous is \boxed{5}.

To determine the value of $ a $ that makes the function

$$
f(n) =
\begin{cases}
4n + 3 & \text{if } n < a, \\
7n - 12 & \text{if } n \ge a,
\end{cases}
$$

continuous, we must ensure that the function has no jumps or breaks at the point where the definition changes, i.e., at $ n = a $.

---

### Step 1: Understand the Concept of Continuity

A function $ f(n) $ is continuous at a point $ n = a $ if:

$$
\lim_{n \to a^-} f(n) = \lim_{n \to a^+} f(n) = f(a)
$$

In this case, the function is defined as two different linear expressions on either side of $ a $. Therefore, the only potential point of discontinuity is at $ n = a $, and we must ensure that the left-hand limit and the right-hand limit are equal to the function value at $ a $.

---

### Step 2: Evaluate the Left-Hand Limit

For $ n < a $, the function is $ f(n) = 4n + 3 $. So, the left-hand limit as $ n \to a^- $ is:

$$
\lim_{n \to a^-} f(n) = 4a + 3
$$

---

### Step 3: Evaluate the Right-Hand Limit

For $ n \ge a $, the function is $ f(n) = 7n - 12 $. So, the right-hand limit as $ n \to a^+ $ is:

$$
\lim_{n \to a^+} f(n) = 7a - 12
$$

---

### Step 4: Set the Limits Equal for Continuity

To ensure continuity at $ n = a $, we set the left-hand limit equal to the right-hand limit:

$$
4a + 3 = 7a - 12
$$

Solving for $ a $:

$$
4a + 3 = 7a - 12 \\
3 + 12 = 7a - 4a \\
15 = 3a \\
a = 5
$$

---

### Step 5: Verify the Continuity at $ a = 5 $

- Left-hand limit: $ 4(5) + 3 = 20 + 3 = 23 $
- Right-hand limit: $ 7(5) - 12 = 35 - 12 = 23 $
- Function value at $ a = 5 $: $ f(5) = 7(5) - 12 = 23 $

All three values are equal, confirming that the function is continuous at $ n = 5 $.

---

### Final Answer

$$
\boxed{5}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of 'a' that makes the function f(n) continuous. The function is defined piecewise: it's 4n + 3 when n is less than a, and 7n - 12 when n is greater than or equal to a. Hmm, continuity means that there are no jumps or breaks in the graph, right? So, at the point where the definition of the function changes, which is at n = a, the two pieces should meet. That means the value of the function from the left (when n approaches a from below) should be equal to the value from the right (when n approaches a from above).

Let me write that down. For continuity at n = a, the limit as n approaches a from the left should equal the limit as n approaches a from the right, and both should equal f(a). Since the function is defined differently on either side of a, I need to set the two expressions equal to each other at n = a.

So, if I plug n = a into both expressions, they should be the same. Let me check that.

First, when n is less than a, the function is 4n + 3. So, approaching a from the left, the value would be 4a + 3.

On the other hand, when n is greater than or equal to a, the function is 7n - 12. So, approaching a from the right, the value would be 7a - 12.

Since the function is continuous at a, these two expressions must be equal. Therefore, I can set up the equation:

4a + 3 = 7a - 12

Now, I need to solve for a. Let me subtract 4a from both sides:

3 = 3a - 12

Then, add 12 to both sides:

15 = 3a

Divide both sides by 3:

a = 5

Wait, let me check that again. If a is 5, then let's verify if the function is continuous there.

For n approaching 5 from the left, f(n) approaches 4*5 + 3 = 20 + 3 = 23.

For n approaching 5 from the right, f(n) approaches 7*5 - 12 = 35 - 12 = 23.

And f(5) itself is 7*5 - 12 = 23. So yes, both sides agree. Therefore, the function is continuous at a = 5.

But wait, let me make sure there isn't any other condition or something I missed. The problem says "the graph of y = f(n) is continuous." So, does that mean continuous everywhere? Well, since the function is defined as two linear functions, which are continuous everywhere on their own. The only potential point of discontinuity is at n = a, so by ensuring continuity at n = a, the entire function is continuous.

Therefore, the answer should be a = 5.

But let me think again. Is there a possibility that the function could have another point where it's discontinuous? For example, if there was a different definition for other values? But according to the problem, the function is defined as 4n + 3 when n < a and 7n - 12 when n ≥ a. So, there's no other points of definition. So, the only point where the function could be discontinuous is at n = a. Therefore, solving for a such that the left-hand limit equals the right-hand limit (and f(a)) is sufficient.

Therefore, my conclusion is that a = 5.

But just to be thorough, let me consider if there's any other way this could be approached. For example, maybe the problem is in terms of integers? Wait, the problem didn't specify that n is an integer. It just says n. So, n is a real number? Or is n an integer? Hmm, the problem didn't specify. But since it's talking about the graph of y = f(n), which is typically a function of a real variable. If n were an integer, the problem might have mentioned it, like "for integer values of n" or something. So, I think n is a real number here.

Therefore, the continuity condition is as I considered before. So, solving 4a + 3 = 7a - 12 gives a = 5.

Alternatively, if n were an integer, then the function would be defined for integer n, and the continuity would be about the function being continuous at integer points. But since the problem says "the graph of y = f(n) is continuous," which usually implies continuity for all real numbers, unless specified otherwise.

But even if n were an integer, the function would be continuous at each integer point. However, since the function is defined piecewise with different expressions for n < a and n ≥ a, the only point where the function could have a discontinuity is at n = a. Therefore, regardless of whether n is real or integer, the condition for continuity at n = a is the same.

Therefore, I think my answer is correct.

Wait, but if n is an integer, then the function is defined for integers. So, for example, if a is an integer, then for n approaching a from the left (i.e., n = a - 1), the function would be 4(a - 1) + 3, and for n = a, it's 7a - 12. So, in that case, the continuity would require that 4(a - 1) + 3 = 7a - 12. Let me check that.

If n is integer, then the left-hand limit as n approaches a from the left would be f(a - 1) = 4(a - 1) + 3 = 4a - 4 + 3 = 4a - 1. The right-hand limit is f(a) = 7a - 12. So, setting them equal: 4a - 1 = 7a - 12. Then, solving: -1 + 12 = 7a - 4a => 11 = 3a => a = 11/3 ≈ 3.666... But if a is supposed to be an integer, then this would not be possible. However, the problem didn't specify that a is an integer.

But the problem says "Find a if the graph of y = f(n) is continuous." If n is an integer, then the graph would consist of discrete points, but the term "continuous" is usually applied to functions defined on real numbers. So, if the problem is in the context of real numbers, then my original answer of a = 5 is correct. If it's in the context of integers, then the problem might be different, but since the problem didn't specify, I think the safe assumption is that n is a real number.

Therefore, the answer is a = 5.

Another way to check: if a = 5, then the function is 4n + 3 for n < 5 and 7n - 12 for n ≥ 5. Let me pick a value less than 5, say n = 4. Then f(4) = 4*4 + 3 = 19. For n = 5, f(5) = 7*5 - 12 = 35 - 12 = 23. If I check the value approaching 5 from the left, say n = 4.9, f(4.9) = 4*(4.9) + 3 = 19.6 + 3 = 22.6. Approaching from the right, n = 5.1, f(5.1) = 7*(5.1) - 12 = 35.7 - 12 = 23.7. Wait, but hold on, if a = 5, then at n = 5, the function is 7n - 12, which is 23. But approaching from the left, the limit is 4*5 + 3 = 23. So, the left limit is 23, the right limit is 7*5 - 12 = 23, and f(5) is 23. So, that's continuous.

But when I checked n = 4.9, which is approaching 5 from the left, I get 22.6, which is approaching 23. Similarly, n = 5.1 gives 23.7, which is approaching 23? Wait, no. Wait, 7*5.1 is 35.7, minus 12 is 23.7. Wait, but if n approaches 5 from the right, then as n approaches 5, 7n - 12 approaches 35 - 12 = 23. So, 23.7 is when n is 5.1, which is further away. So, as n approaches 5 from the right, the value approaches 23. Similarly, approaching from the left, approaching 5, the value approaches 23. Therefore, the function is continuous at 5.

So, my initial calculation was correct. Therefore, a = 5 is the correct answer.

I think I've checked it thoroughly. So, I think the answer is 5.

**Final Answer**
The value of $ a $ that makes the graph of $ y = f(n) $ continuous is \boxed{5}.

To determine the value of $ a $ that makes the function

$$
f(n) =
\begin{cases}
4n + 3 & \text{if } n < a, \\
7n - 12 & \text{if } n \ge a,
\end{cases}
$$

continuous, we must ensure that the function has no jumps or breaks at the point where the definition changes, i.e., at $ n = a $.

---

### Step 1: Understand the Concept of Continuity

A function $ f(n) $ is continuous at a point $ n = a $ if:

$$
\lim_{n \to a^-} f(n) = \lim_{n \to a^+} f(n) = f(a)
$$

In this case, the function is defined as two different linear expressions on either side of $ a $. Therefore, the only potential point of discontinuity is at $ n = a $, and we must ensure that the left-hand limit and the right-hand limit are equal to the function value at $ a $.

---

### Step 2: Evaluate the Left-Hand Limit

For $ n < a $, the function is $ f(n) = 4n + 3 $. So, the left-hand limit as $ n \to a^- $ is:

$$
\lim_{n \to a^-} f(n) = 4a + 3
$$

---

### Step 3: Evaluate the Right-Hand Limit

For $ n \ge a $, the function is $ f(n) = 7n - 12 $. So, the right-hand limit as $ n \to a^+ $ is:

$$
\lim_{n \to a^+} f(n) = 7a - 12
$$

---

### Step 4: Set the Limits Equal for Continuity

To ensure continuity at $ n = a $, we set the left-hand limit equal to the right-hand limit:

$$
4a + 3 = 7a - 12
$$

Solving for $ a $:

$$
4a + 3 = 7a - 12 \\
3 + 12 = 7a - 4a \\
15 = 3a \\
a = 5
$$

---

### Step 5: Verify the Continuity at $ a = 5 $

- Left-hand limit: $ 4(5) + 3 = 20 + 3 = 23 $
- Right-hand limit: $ 7(5) - 12 = 35 - 12 = 23 $
- Function value at $ a = 5 $: $ f(5) = 7(5) - 12 = 23 $

All three values are equal, confirming that the function is continuous at $ n = 5 $.

---

### Final Answer

$$
\boxed{5}
$$

Solution: In order for the graph to be continuous, the two pieces of the function must meet at $n=a$. In order for this to happen, we know that $4a+3=7a-12$. Solving for $a$, we find that $a=\frac{15}{3}=\boxed{5}$.


Explanation:
Okay, so I need to compute the binomial coefficient $\dbinom{31}{28}$. Hmm, let me remember how binomial coefficients work. I think they're used in combinations, right? Like, the number of ways to choose 28 items out of 31 without considering the order. But wait, there's a formula for that. Let me recall... I think it's $\dbinom{n}{k} = \frac{n!}{k!(n - k)!}$. Yeah, that sounds right. So, applying that formula here, it should be $\frac{31!}{28!(31 - 28)!}$. Let me check the subtraction in the denominator: 31 minus 28 is 3. So, the denominator becomes 28! times 3!.

But wait, factorials can get really big. Maybe there's a way to simplify this before calculating? Let me think. Since 31! divided by 28! would be 31 × 30 × 29 × 28! divided by 28!... Oh, right! The 28! cancels out from numerator and denominator. So that simplifies to 31 × 30 × 29 divided by 3!.

So, let me write that down: $\dbinom{31}{28} = \frac{31 \times 30 \times 29}{3 \times 2 \times 1}$. Let me compute the numerator first. 31 multiplied by 30 is... 31*30. Let me do 30*30 which is 900, plus 1*30 which is 30, so total 930. Then multiply that by 29. Hmm, 930*29. Let me break that down. 930*30 would be 27,900, so subtract 930 to get 930*29. So 27,900 - 930 = 26,970. So the numerator is 26,970.

Now the denominator is 3! which is 6. So, divide 26,970 by 6. Let me do that division. 26,970 divided by 6. Let me see, 6 goes into 26 four times (4*6=24), remainder 2. Bring down the 9 to make 29. 6 into 29 is 4 times (4*6=24), remainder 5. Bring down the 7 to make 57. 6 into 57 is 9 times (9*6=54), remainder 3. Bring down the 0 to make 30. 6 into 30 is 5 times. So putting that all together: 4, 4, 9, 5. Wait, let me check again step by step.

Wait, maybe I should do it another way. Let me divide 26,970 by 6. Let me divide each part:

First, 26,970 divided by 6. Let me divide 26,970 by 6:

Divide 26,970 by 6:

6 × 4,000 = 24,000. Subtract 24,000 from 26,970: 2,970.

Now divide 2,970 by 6. 6 × 400 = 2,400. Subtract: 2,970 - 2,400 = 570.

Divide 570 by 6: 6 × 95 = 570. So total is 4,000 + 400 + 95 = 4,495.

So, 26,970 divided by 6 is 4,495. Therefore, $\dbinom{31}{28} = 4,495$.

Wait, but let me verify this because sometimes when I do division step by step, I might make a mistake. Let me check with another method. Let me compute 26,970 divided by 6.

Alternatively, 26,970 divided by 6. Let me divide 26,970 by 6:

First, divide 26,970 by 2: 13,485. Then divide by 3: 13,485 ÷ 3. Let me compute that. 3 × 4,000 = 12,000. Subtract 12,000 from 13,485: 1,485. 3 × 400 = 1,200. Subtract: 1,485 - 1,200 = 285. 3 × 95 = 285. So total is 4,000 + 400 + 95 = 4,495. Yep, same result. So that seems correct.

But just to be thorough, maybe I can check using another approach. Let me recall that $\dbinom{n}{k} = \dbinom{n}{n - k}$. So, $\dbinom{31}{28} = \dbinom{31}{3}$. Maybe computing $\dbinom{31}{3}$ is easier? Let me try that.

$\dbinom{31}{3} = \frac{31 \times 30 \times 29}{3 \times 2 \times 1}$, which is exactly the same as before. So that's 31*30*29 / 6. Which is the same calculation. So, same result. So that's reassuring.

Alternatively, maybe I can compute $\dbinom{31}{3}$ step by step. Let me compute 31 choose 3.

First, 31 × 30 = 930, as before. Then 930 × 29. Wait, that's the same numerator. Then divide by 6. So same thing. So, same answer.

Alternatively, maybe I can compute 31 × 30 × 29 divided by 6. Let me see if I can simplify before multiplying. For example, 31 divided by something? Well, 31 is a prime number, so maybe not. Let me see. Let me write the numerator as 31 × 30 × 29. The denominator is 6, which is 2 × 3. Let me see if I can cancel factors.

So, 30 in the numerator is 2 × 3 × 5. So, 30 divided by 6 is 5. So, if I divide 30 by 6, that would be 5. So, then the expression becomes 31 × 5 × 29. Let me check that.

Original expression: (31 × 30 × 29) / 6. Since 30 / 6 = 5, so yes, that simplifies to 31 × 5 × 29.

So, compute 31 × 5 first. 31 × 5 is 155. Then multiply by 29. So, 155 × 29. Let me compute that.

Breaking down 155 × 29: 155 × 30 = 4,650. Subtract 155 to get 155 × 29. So, 4,650 - 155 = 4,495.

Same result! So that's another way to compute it, and it gives the same answer. So that's good.

Therefore, regardless of the method, I get 4,495.

But just to be absolutely sure, maybe I can check with a calculator? Wait, since I don't have a calculator here, but maybe I can verify with smaller numbers. Let me test with a known value. For example, $\dbinom{5}{2} = 10$. Let me apply the same method.

$\dbinom{5}{2} = \frac{5!}{2!3!} = \frac{5 × 4}{2 × 1} = 10$. Correct.

Another example: $\dbinom{10}{3} = 120$. Let me compute that. 10 × 9 × 8 / 6 = 720 / 6 = 120. Correct.

So, the method works. Therefore, applying the same method to $\dbinom{31}{28}$ gives 4,495.

Alternatively, maybe I can use Pascal's triangle or some recursive formula? But that might take longer.

Alternatively, think about the multiplicative formula for combinations. The formula is $\dbinom{n}{k} = \frac{n(n - 1)(n - 2)...(n - k + 1)}{k!}$. So, for $\dbinom{31}{28}$, that would be $\frac{31 × 30 × 29 × ... × (31 - 28 + 1)}{28!}$. Wait, but 31 - 28 + 1 is 4. So, the numerator would be 31 × 30 × 29 × 28! divided by 28! which cancels out, giving 31 × 30 × 29 / 3! again. So same as before.

Therefore, regardless of the approach, the answer is 4,495.

Wait, but just to make sure, let me check if 31 choose 28 is indeed equal to 31 choose 3. Since 31 - 28 = 3. So, yes, $\dbinom{31}{28} = \dbinom{31}{3}$. Let me check with another example. For instance, $\dbinom{5}{2} = \dbinom{5}{3} = 10$. Correct. So, that's a valid property. Therefore, computing $\dbinom{31}{3}$ is equivalent.

So, if I compute $\dbinom{31}{3}$, which is 31 × 30 × 29 / 6, which we did earlier, and got 4,495.

Alternatively, maybe I can compute 31 × 30 first, which is 930, then multiply by 29. Let me do that again.

930 × 29. Let me think of 29 as 30 - 1. So, 930 × 30 = 27,900. Then subtract 930 × 1 = 930. So, 27,900 - 930 = 26,970. Then divide by 6. Which is 26,970 / 6 = 4,495. Yep.

Alternatively, maybe I can compute 26,970 divided by 6 by breaking it down. 26,970 divided by 6. Let me divide each digit:

6 into 2 is 0, 6 into 26 is 4 (4×6=24), remainder 2. Bring down 9: 29. 6 into 29 is 4 (4×6=24), remainder 5. Bring down 7: 57. 6 into 57 is 9 (9×6=54), remainder 3. Bring down 0: 30. 6 into 30 is 5. So the quotient is 4,495. Yep, same as before.

So, all methods lead to 4,495. Therefore, I think that's the correct answer.

But just to be 100% sure, maybe I can check with another approach. Let me think.

Suppose I compute $\dbinom{31}{28}$ using the multiplicative formula step by step.

The multiplicative formula is $\dbinom{n}{k} = \frac{n}{k} \times \frac{n - 1}{k - 1} \times \cdots \times \frac{n - k + 1}{1}$.

But for $\dbinom{31}{28}$, that would be $\frac{31}{28} \times \frac{30}{27} \times \frac{29}{26} \times \cdots$? Wait, no, that seems more complicated. Maybe not the best approach.

Alternatively, since $\dbinom{31}{28} = \dbinom{31}{3}$, let me compute $\dbinom{31}{3}$ using the multiplicative formula.

The formula is $\dbinom{n}{k} = \frac{n \times (n - 1) \times (n - 2)}{k \times (k - 1) \times (k - 2)}$.

So, for $\dbinom{31}{3}$, that would be $\frac{31 × 30 × 29}{3 × 2 × 1}$, which is the same as before. So, again, same calculation.

Alternatively, maybe compute step by step:

First term: 31 / 3 = 10.333... Hmm, but fractions might complicate.

Alternatively, compute 31 × 30 = 930, then 930 × 29 = 26,970, then divide by 6. Which is 4,495.

Alternatively, think of 31 × 30 × 29 divided by 6. Let me factor all numbers:

31 is prime. 30 is 2 × 3 × 5. 29 is prime. So numerator is 31 × 2 × 3 × 5 × 29. Denominator is 2 × 3. So cancel out 2 and 3, leaving 31 × 5 × 29. Which is 31 × 5 = 155, 155 × 29. Which is 155 × 30 - 155 = 4,650 - 155 = 4,495.

Same result.

Therefore, I think I can be confident that the answer is 4,495.

But just to make sure, let me check with another example. Let me compute $\dbinom{10}{3}$ using this method.

$\dbinom{10}{3} = \frac{10 × 9 × 8}{3 × 2 × 1} = \frac{720}{6} = 120$. Which is correct.

Another example: $\dbinom{5}{2} = \frac{5 × 4}{2 × 1} = 10$. Correct.

So, the method works. Therefore, applying it to $\dbinom{31}{28}$ gives 4,495.

Alternatively, maybe I can use the recursive formula: $\dbinom{n}{k} = \dbinom{n - 1}{k - 1} + \dbinom{n - 1}{k}$. But computing $\dbinom{31}{28}$ recursively would take a lot of steps, so maybe not practical here.

But just for fun, let me try to compute $\dbinom{31}{28}$ using recursion.

But since $\dbinom{31}{28} = \dbinom{30}{27} + \dbinom{30}{28}$. Then $\dbinom{30}{27} = \dbinom{29}{26} + \dbinom{29}{27}$, and so on. This seems too tedious.

Alternatively, maybe use known values. But without a table, it's not helpful.

Alternatively, think about the value of $\dbinom{31}{3}$. Let me think of 31 choose 3.

I know that $\dbinom{n}{3} = \frac{n(n - 1)(n - 2)}{6}$. So, plugging in n = 31, we get $\frac{31 × 30 × 29}{6}$. Which is exactly what we did.

Alternatively, maybe approximate? Well, 31 × 30 × 29 is roughly 30 × 30 × 30 = 27,000. Divided by 6 is 4,500. Which is very close to our exact answer of 4,495. So that approximation makes sense.

Therefore, given all these cross-checks and methods, I can be confident that the answer is 4,495.

**Final Answer**
\boxed{4495}

To compute the binomial coefficient $\dbinom{31}{28}$, we use the standard formula for combinations:

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

---

### Step 1: Apply the formula

We are given $n = 31$ and $k = 28$, so:

$$
\dbinom{31}{28} = \frac{31!}{28!(31 - 28)!} = \frac{31!}{28! \cdot 3!}
$$

---

### Step 2: Simplify the factorials

Note that:

$$
\frac{31!}{28!} = 31 \times 30 \times 29
$$

So the expression becomes:

$$
\dbinom{31}{28} = \frac{31 \times 30 \times 29}{3!} = \frac{31 \times 30 \times 29}{6}
$$

---

### Step 3: Compute the numerator

$$
31 \times 30 = 930
$$
$$
930 \times 29 = 26,970
$$

---

### Step 4: Divide by the denominator

$$
\frac{26,970}{6} = 4,495
$$

---

### Step 5: Final Answer

$$
\boxed{4495}
$$

[Assistant]
Explanation:
Okay, so I need to compute the binomial coefficient $\dbinom{31}{28}$. Hmm, let me remember how binomial coefficients work. I think they're used in combinations, right? Like, the number of ways to choose 28 items out of 31 without considering the order. But wait, there's a formula for that. Let me recall... I think it's $\dbinom{n}{k} = \frac{n!}{k!(n - k)!}$. Yeah, that sounds right. So, applying that formula here, it should be $\frac{31!}{28!(31 - 28)!}$. Let me check the subtraction in the denominator: 31 minus 28 is 3. So, the denominator becomes 28! times 3!.

But wait, factorials can get really big. Maybe there's a way to simplify this before calculating? Let me think. Since 31! divided by 28! would be 31 × 30 × 29 × 28! divided by 28!... Oh, right! The 28! cancels out from numerator and denominator. So that simplifies to 31 × 30 × 29 divided by 3!.

So, let me write that down: $\dbinom{31}{28} = \frac{31 \times 30 \times 29}{3 \times 2 \times 1}$. Let me compute the numerator first. 31 multiplied by 30 is... 31*30. Let me do 30*30 which is 900, plus 1*30 which is 30, so total 930. Then multiply that by 29. Hmm, 930*29. Let me break that down. 930*30 would be 27,900, so subtract 930 to get 930*29. So 27,900 - 930 = 26,970. So the numerator is 26,970.

Now the denominator is 3! which is 6. So, divide 26,970 by 6. Let me do that division. 26,970 divided by 6. Let me see, 6 goes into 26 four times (4*6=24), remainder 2. Bring down the 9 to make 29. 6 into 29 is 4 times (4*6=24), remainder 5. Bring down the 7 to make 57. 6 into 57 is 9 times (9*6=54), remainder 3. Bring down the 0 to make 30. 6 into 30 is 5 times. So putting that all together: 4, 4, 9, 5. Wait, let me check again step by step.

Wait, maybe I should do it another way. Let me divide 26,970 by 6. Let me divide each part:

First, 26,970 divided by 6. Let me divide 26,970 by 6:

Divide 26,970 by 6:

6 × 4,000 = 24,000. Subtract 24,000 from 26,970: 2,970.

Now divide 2,970 by 6. 6 × 400 = 2,400. Subtract: 2,970 - 2,400 = 570.

Divide 570 by 6: 6 × 95 = 570. So total is 4,000 + 400 + 95 = 4,495.

So, 26,970 divided by 6 is 4,495. Therefore, $\dbinom{31}{28} = 4,495$.

Wait, but let me verify this because sometimes when I do division step by step, I might make a mistake. Let me check with another method. Let me compute 26,970 divided by 6.

Alternatively, 26,970 divided by 6. Let me divide 26,970 by 6:

First, divide 26,970 by 2: 13,485. Then divide by 3: 13,485 ÷ 3. Let me compute that. 3 × 4,000 = 12,000. Subtract 12,000 from 13,485: 1,485. 3 × 400 = 1,200. Subtract: 1,485 - 1,200 = 285. 3 × 95 = 285. So total is 4,000 + 400 + 95 = 4,495. Yep, same result. So that seems correct.

But just to be thorough, maybe I can check using another approach. Let me recall that $\dbinom{n}{k} = \dbinom{n}{n - k}$. So, $\dbinom{31}{28} = \dbinom{31}{3}$. Maybe computing $\dbinom{31}{3}$ is easier? Let me try that.

$\dbinom{31}{3} = \frac{31 \times 30 \times 29}{3 \times 2 \times 1}$, which is exactly the same as before. So that's 31*30*29 / 6. Which is the same calculation. So, same result. So that's reassuring.

Alternatively, maybe I can compute $\dbinom{31}{3}$ step by step. Let me compute 31 choose 3.

First, 31 × 30 = 930, as before. Then 930 × 29. Wait, that's the same numerator. Then divide by 6. So same thing. So, same answer.

Alternatively, maybe I can compute 31 × 30 × 29 divided by 6. Let me see if I can simplify before multiplying. For example, 31 divided by something? Well, 31 is a prime number, so maybe not. Let me see. Let me write the numerator as 31 × 30 × 29. The denominator is 6, which is 2 × 3. Let me see if I can cancel factors.

So, 30 in the numerator is 2 × 3 × 5. So, 30 divided by 6 is 5. So, if I divide 30 by 6, that would be 5. So, then the expression becomes 31 × 5 × 29. Let me check that.

Original expression: (31 × 30 × 29) / 6. Since 30 / 6 = 5, so yes, that simplifies to 31 × 5 × 29.

So, compute 31 × 5 first. 31 × 5 is 155. Then multiply by 29. So, 155 × 29. Let me compute that.

Breaking down 155 × 29: 155 × 30 = 4,650. Subtract 155 to get 155 × 29. So, 4,650 - 155 = 4,495.

Same result! So that's another way to compute it, and it gives the same answer. So that's good.

Therefore, regardless of the method, I get 4,495.

But just to be absolutely sure, maybe I can check with a calculator? Wait, since I don't have a calculator here, but maybe I can verify with smaller numbers. Let me test with a known value. For example, $\dbinom{5}{2} = 10$. Let me apply the same method.

$\dbinom{5}{2} = \frac{5!}{2!3!} = \frac{5 × 4}{2 × 1} = 10$. Correct.

Another example: $\dbinom{10}{3} = 120$. Let me compute that. 10 × 9 × 8 / 6 = 720 / 6 = 120. Correct.

So, the method works. Therefore, applying the same method to $\dbinom{31}{28}$ gives 4,495.

Alternatively, maybe I can use Pascal's triangle or some recursive formula? But that might take longer.

Alternatively, think about the multiplicative formula for combinations. The formula is $\dbinom{n}{k} = \frac{n(n - 1)(n - 2)...(n - k + 1)}{k!}$. So, for $\dbinom{31}{28}$, that would be $\frac{31 × 30 × 29 × ... × (31 - 28 + 1)}{28!}$. Wait, but 31 - 28 + 1 is 4. So, the numerator would be 31 × 30 × 29 × 28! divided by 28! which cancels out, giving 31 × 30 × 29 / 3! again. So same as before.

Therefore, regardless of the approach, the answer is 4,495.

Wait, but just to make sure, let me check if 31 choose 28 is indeed equal to 31 choose 3. Since 31 - 28 = 3. So, yes, $\dbinom{31}{28} = \dbinom{31}{3}$. Let me check with another example. For instance, $\dbinom{5}{2} = \dbinom{5}{3} = 10$. Correct. So, that's a valid property. Therefore, computing $\dbinom{31}{3}$ is equivalent.

So, if I compute $\dbinom{31}{3}$, which is 31 × 30 × 29 / 6, which we did earlier, and got 4,495.

Alternatively, maybe I can compute 31 × 30 first, which is 930, then multiply by 29. Let me do that again.

930 × 29. Let me think of 29 as 30 - 1. So, 930 × 30 = 27,900. Then subtract 930 × 1 = 930. So, 27,900 - 930 = 26,970. Then divide by 6. Which is 26,970 / 6 = 4,495. Yep.

Alternatively, maybe I can compute 26,970 divided by 6 by breaking it down. 26,970 divided by 6. Let me divide each digit:

6 into 2 is 0, 6 into 26 is 4 (4×6=24), remainder 2. Bring down 9: 29. 6 into 29 is 4 (4×6=24), remainder 5. Bring down 7: 57. 6 into 57 is 9 (9×6=54), remainder 3. Bring down 0: 30. 6 into 30 is 5. So the quotient is 4,495. Yep, same as before.

So, all methods lead to 4,495. Therefore, I think that's the correct answer.

But just to be 100% sure, maybe I can check with another approach. Let me think.

Suppose I compute $\dbinom{31}{28}$ using the multiplicative formula step by step.

The multiplicative formula is $\dbinom{n}{k} = \frac{n}{k} \times \frac{n - 1}{k - 1} \times \cdots \times \frac{n - k + 1}{1}$.

But for $\dbinom{31}{28}$, that would be $\frac{31}{28} \times \frac{30}{27} \times \frac{29}{26} \times \cdots$? Wait, no, that seems more complicated. Maybe not the best approach.

Alternatively, since $\dbinom{31}{28} = \dbinom{31}{3}$, let me compute $\dbinom{31}{3}$ using the multiplicative formula.

The formula is $\dbinom{n}{k} = \frac{n \times (n - 1) \times (n - 2)}{k \times (k - 1) \times (k - 2)}$.

So, for $\dbinom{31}{3}$, that would be $\frac{31 × 30 × 29}{3 × 2 × 1}$, which is the same as before. So, again, same calculation.

Alternatively, maybe compute step by step:

First term: 31 / 3 = 10.333... Hmm, but fractions might complicate.

Alternatively, compute 31 × 30 = 930, then 930 × 29 = 26,970, then divide by 6. Which is 4,495.

Alternatively, think of 31 × 30 × 29 divided by 6. Let me factor all numbers:

31 is prime. 30 is 2 × 3 × 5. 29 is prime. So numerator is 31 × 2 × 3 × 5 × 29. Denominator is 2 × 3. So cancel out 2 and 3, leaving 31 × 5 × 29. Which is 31 × 5 = 155, 155 × 29. Which is 155 × 30 - 155 = 4,650 - 155 = 4,495.

Same result.

Therefore, I think I can be confident that the answer is 4,495.

But just to make sure, let me check with another example. Let me compute $\dbinom{10}{3}$ using this method.

$\dbinom{10}{3} = \frac{10 × 9 × 8}{3 × 2 × 1} = \frac{720}{6} = 120$. Which is correct.

Another example: $\dbinom{5}{2} = \frac{5 × 4}{2 × 1} = 10$. Correct.

So, the method works. Therefore, applying it to $\dbinom{31}{28}$ gives 4,495.

Alternatively, maybe I can use the recursive formula: $\dbinom{n}{k} = \dbinom{n - 1}{k - 1} + \dbinom{n - 1}{k}$. But computing $\dbinom{31}{28}$ recursively would take a lot of steps, so maybe not practical here.

But just for fun, let me try to compute $\dbinom{31}{28}$ using recursion.

But since $\dbinom{31}{28} = \dbinom{30}{27} + \dbinom{30}{28}$. Then $\dbinom{30}{27} = \dbinom{29}{26} + \dbinom{29}{27}$, and so on. This seems too tedious.

Alternatively, maybe use known values. But without a table, it's not helpful.

Alternatively, think about the value of $\dbinom{31}{3}$. Let me think of 31 choose 3.

I know that $\dbinom{n}{3} = \frac{n(n - 1)(n - 2)}{6}$. So, plugging in n = 31, we get $\frac{31 × 30 × 29}{6}$. Which is exactly what we did.

Alternatively, maybe approximate? Well, 31 × 30 × 29 is roughly 30 × 30 × 30 = 27,000. Divided by 6 is 4,500. Which is very close to our exact answer of 4,495. So that approximation makes sense.

Therefore, given all these cross-checks and methods, I can be confident that the answer is 4,495.

**Final Answer**
\boxed{4495}

To compute the binomial coefficient $\dbinom{31}{28}$, we use the standard formula for combinations:

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

---

### Step 1: Apply the formula

We are given $n = 31$ and $k = 28$, so:

$$
\dbinom{31}{28} = \frac{31!}{28!(31 - 28)!} = \frac{31!}{28! \cdot 3!}
$$

---

### Step 2: Simplify the factorials

Note that:

$$
\frac{31!}{28!} = 31 \times 30 \times 29
$$

So the expression becomes:

$$
\dbinom{31}{28} = \frac{31 \times 30 \times 29}{3!} = \frac{31 \times 30 \times 29}{6}
$$

---

### Step 3: Compute the numerator

$$
31 \times 30 = 930
$$
$$
930 \times 29 = 26,970
$$

---

### Step 4: Divide by the denominator

$$
\frac{26,970}{6} = 4,495
$$

---

### Step 5: Final Answer

$$
\boxed{4495}
$$

Solution:
\begin{align*}
\dbinom{31}{28} &= \dbinom{31}{3} \\
&= \dfrac{31!}{28!3!} \\
&= \dfrac{31\times 30\times 29}{3\times 2\times 1} \\
&= 31 \times \dfrac{30}{3\times 2\times 1} \times 29 \\
&= 31\times 5\times 29 \\
&= \boxed{4495}.
\end{align*}