Explanation:
Okay, so I need to find the value of the product sin20° sin40° sin60° sin80° sin100° sin120° sin140° sin160°. Hmm, that's a lot of sine terms multiplied together. Let me think about how to approach this.

First, I remember that there are some trigonometric identities that can help simplify products of sines. Maybe I can pair up the terms or use some product-to-sum formulas? But with eight terms, that might get complicated. Let me see if there's a pattern or symmetry here.

Looking at the angles: 20°, 40°, 60°, 80°, 100°, 120°, 140°, 160°. These angles are all multiples of 20°, starting from 20° up to 160°, increasing by 20° each time. Wait, 20°, 40°, 60°, 80°, then 100°, which is 180° - 80°, 120° is 180° - 60°, 140° is 180° - 40°, and 160° is 180° - 20°. So, actually, the angles from 100° to 160° are supplementary angles to the first four angles.

I remember that sin(180° - x) = sinx. So, sin100° = sin80°, sin120° = sin60°, sin140° = sin40°, sin160° = sin20°. Therefore, the product can be rewritten as:

[sin20° sin40° sin60° sin80°] × [sin80° sin60° sin40° sin20°]

Which is [sin20° sin40° sin60° sin80°]^2. So, the original product is the square of the product of sin20°, sin40°, sin60°, and sin80°. That simplifies things a bit. So now I just need to compute [sin20° sin40° sin60° sin80°]^2.

But maybe there's a better way. Let me check if I can compute the product sin20° sin40° sin60° sin80° first. Let me recall some identities. There's a formula for the product of sines in an arithmetic sequence? Or maybe using complex numbers? Or perhaps using multiple-angle identities?

Alternatively, I remember that there's a formula for the product of sines of angles in arithmetic progression. But I don't remember exactly. Let me think. Alternatively, maybe I can pair terms. For example, sin20° sin80° and sin40° sin60°?

Let me try that. Let me compute sin20° sin80° first. Using the identity: sinA sinB = [cos(A - B) - cos(A + B)] / 2.

So, sin20° sin80° = [cos(20° - 80°) - cos(20° + 80°)] / 2 = [cos(-60°) - cos(100°)] / 2. Since cos(-x) = cosx, this becomes [cos60° - cos100°]/2.

Similarly, cos60° is 0.5, so that term is 0.5 - cos100°, divided by 2.

Similarly, let me compute sin40° sin60°. Using the same identity:

sin40° sin60° = [cos(40° - 60°) - cos(40° + 60°)] / 2 = [cos(-20°) - cos100°]/2 = [cos20° - cos100°]/2.

So, putting it all together, the product sin20° sin40° sin60° sin80° is equal to [ (cos60° - cos100°)/2 ] × [ (cos20° - cos100°)/2 ].

Hmm, that seems a bit messy, but maybe I can compute it. Let me write that down:

Product = [ (cos60° - cos100°)(cos20° - cos100°) ] / 4.

Let me compute the numerator:

(cos60° - cos100°)(cos20° - cos100°) = cos60° cos20° - cos60° cos100° - cos100° cos20° + cos²100°.

Hmm, this seems complicated. Maybe there's another approach. Let me think again.

Alternatively, maybe I can use the identity for sinθ sin(60° - θ) sin(60° + θ) = (sin3θ)/4. Wait, is that a valid identity? Let me check.

Yes, I think there is an identity that says sinθ sin(60° - θ) sin(60° + θ) = (sin3θ)/4. Let me verify with θ = 20°, then sin20° sin40° sin80° = sin60° / 4? Wait, sin3θ would be sin60°, which is √3/2. So, according to the identity, sin20° sin40° sin80° = (√3/2)/4 = √3/8? Let me check with actual values.

But maybe I can use this identity. Let me check if that's correct. Let me recall that formula.

Yes, the identity is: sinθ sin(60° - θ) sin(60° + θ) = (sin3θ)/4.

So, if I take θ = 20°, then sin20° sin40° sin80° = sin60° / 4.

So, sin20° sin40° sin80° = (√3/2)/4 = √3/8.

But in our case, we have sin20° sin40° sin60° sin80°. So, if I take the product sin20° sin40° sin80°, which is √3/8, and then multiply by sin60°, which is √3/2. Therefore, the product sin20° sin40° sin60° sin80° = (√3/8) × (√3/2) = (3/16).

Wait, that seems promising. Let me check again.

So, using the identity sinθ sin(60° - θ) sin(60° + θ) = sin3θ / 4.

So, if θ = 20°, then sin20° sin(60° - 20°) sin(60° + 20°) = sin20° sin40° sin80° = sin(60°)/4 = (√3/2)/4 = √3/8.

Therefore, sin20° sin40° sin80° = √3/8. Then, multiplying by sin60°, which is √3/2, gives us √3/8 × √3/2 = (3)/16.

So, sin20° sin40° sin60° sin80° = 3/16.

Therefore, the original product is [sin20° sin40° sin60° sin80°]^2 = (3/16)^2 = 9/256.

Wait, but hold on. Let me check if that's correct. Because the original product was sin20° sin40° sin60° sin80° sin100° sin120° sin140° sin160°, which we converted to [sin20° sin40° sin60° sin80°]^2.

But according to the above, that square would be (3/16)^2 = 9/256. But let me check if that's correct.

Alternatively, maybe I made a mistake in the identity. Let me verify the identity again.

Let me compute sinθ sin(60° - θ) sin(60° + θ). Let me expand it step by step.

First, compute sin(60° - θ) sin(60° + θ). Using the identity sin(A - B) sin(A + B) = sin²A - sin²B. Wait, no, that's cos(A - B) cos(A + B) = cos²A - sin²B? Wait, let me recall:

Actually, sin(A - B) sin(A + B) = sin²A - sin²B. Wait, let me check with A = 60°, B = θ. Then sin(60° - θ) sin(60° + θ) = sin²60° - sin²θ.

Yes, that's correct. Because sin(A - B) sin(A + B) = sin²A - sin²B.

Therefore, sinθ [sin²60° - sin²θ] = sinθ [ (3/4) - sin²θ ].

But according to the identity, this should equal sin3θ / 4. Let me check for θ = 20°, sin3θ = sin60° = √3/2.

So, let's compute sinθ [ (3/4) - sin²θ ] for θ = 20°.

First, sin20° ≈ 0.3420, sin²20° ≈ 0.11699. Then, 3/4 - 0.11699 ≈ 0.75 - 0.11699 ≈ 0.6330. Multiply by sin20° ≈ 0.3420 * 0.6330 ≈ 0.2166.

On the other hand, sin60° / 4 ≈ (√3/2)/4 ≈ 0.4330 / 4 ≈ 0.10825. Wait, that's not matching. Wait, so there's a discrepancy here.

Wait, so according to my calculation, sinθ [sin²60° - sin²θ] ≈ 0.2166, but according to the identity, it should be sin3θ / 4 ≈ 0.10825. So, there's a factor of 2 difference. Therefore, my previous assumption about the identity is wrong.

Hmm, so maybe the correct identity is different? Let me check again.

Alternatively, maybe I need to use another identity. Let me check the product sinθ sin(60° - θ) sin(60° + θ). Let me compute it using another method.

Let me use complex numbers or another identity. Let me recall that sin3θ = 3sinθ - 4sin³θ. But how does that relate?

Alternatively, let me compute sinθ sin(60° - θ) sin(60° + θ). Let me first compute sin(60° - θ) sin(60° + θ) as before, which is sin²60° - sin²θ. Then multiply by sinθ:

sinθ [sin²60° - sin²θ] = sinθ * (3/4 - sin²θ).

Alternatively, maybe express sinθ * sin²60° - sin³θ.

But how does that relate to sin3θ? Let me compute sin3θ.

sin3θ = 3sinθ - 4sin³θ. Therefore, 3sinθ - sin3θ = 4sin³θ.

But I have sinθ*(3/4) - sin³θ = (3/4) sinθ - sin³θ. Which is (3/4) sinθ - sin³θ.

Comparing with 3sinθ - sin3θ = 4sin³θ.

So, if I multiply both sides by 1/4: (3/4) sinθ - (1/4) sin3θ = sin³θ.

Therefore, sinθ*(3/4) - sin³θ = (1/4) sin3θ.

Therefore, sinθ [sin²60° - sin²θ] = (1/4) sin3θ.

Therefore, the identity is sinθ sin(60° - θ) sin(60° + θ) = (1/4) sin3θ.

So, earlier, I had forgotten the 1/4 factor. Therefore, for θ = 20°, sin20° sin40° sin80° = (1/4) sin60° = (1/4)(√3/2) = √3/8. Which matches the previous result.

So, my initial calculation was correct. Then, when I calculated the approximate value, I must have made a mistake. Let me check again.

For θ = 20°, sinθ [sin²60° - sin²θ] = sin20° [ ( (√3/2)^2 - sin²20° ) ] = sin20° [ 3/4 - sin²20° ] ≈ 0.3420 * [ 0.75 - 0.11699 ] ≈ 0.3420 * 0.633 ≈ 0.2166.

On the other hand, (1/4) sin60° ≈ (1/4)(0.8660) ≈ 0.2165. Which matches! So, my previous calculation was correct. So, the identity is correct.

Therefore, sin20° sin40° sin80° = √3/8. Then, multiplying by sin60°, which is √3/2, gives us √3/8 * √3/2 = (3)/16.

Therefore, sin20° sin40° sin60° sin80° = 3/16.

Therefore, the original product is [3/16]^2 = 9/256.

Wait, but let me check again. The original product was sin20° sin40° sin60° sin80° sin100° sin120° sin140° sin160°, which we converted to [sin20° sin40° sin60° sin80°]^2.

But let me confirm that. Since sin100° = sin80°, sin120° = sin60°, sin140° = sin40°, sin160° = sin20°, so the product is [sin20° sin40° sin60° sin80°] × [sin80° sin60° sin40° sin20°] = [sin20° sin40° sin60° sin80°]^2.

Yes, that's correct. Therefore, if sin20° sin40° sin60° sin80° = 3/16, then the square is 9/256.

But let me check with another approach to make sure.

Alternatively, maybe I can use the formula for the product of sines. There's a general formula for the product of sines of the form sinθ sin(60° - θ) sin(60° + θ) = (sin3θ)/4.

But in our case, we have more terms. Wait, but we already used that identity for part of the product.

Alternatively, maybe I can use the identity for multiple angles. Let me think.

Alternatively, since we have angles that are multiples of 20°, maybe we can relate them to roots of unity or something?

Alternatively, let me consider that the product of sines can be related to the formula for sin(nθ) or something.

Alternatively, let me recall that the product of sin(kπ/(2n)) for k=1 to n-1 is something... Wait, but here we have degrees, not radians. Let me convert to radians.

But maybe that's overcomplicating. Let me think again.

Alternatively, since we have eight terms, maybe we can pair them as follows:

sin20° sin160°, sin40° sin140°, sin60° sin120°, sin80° sin100°.

But since sin160° = sin20°, sin140° = sin40°, sin120° = sin60°, sin100° = sin80°, so each pair is [sin20° sin20°], [sin40° sin40°], [sin60° sin60°], [sin80° sin80°]. Wait, no. Wait, original product is sin20° sin40° sin60° sin80° sin100° sin120° sin140° sin160°, which is equal to [sin20° sin160°][sin40° sin140°][sin60° sin120°][sin80° sin100°].

Each pair is sinθ sin(180° - θ) = sinθ sinθ = sin²θ. Wait, no. Wait, sinθ sin(180° - θ) = sinθ sinθ? Wait, sin(180° - θ) = sinθ, so each pair is sinθ * sinθ = sin²θ. Therefore, the entire product is [sin²20° sin²40° sin²60° sin²80°].

Wait, but that contradicts my previous conclusion. Wait, hold on. Wait, if I pair sin20° with sin160°, which is sin20°, so sin20° * sin160° = sin²20°, similarly for others. Therefore, the entire product is [sin²20° sin²40° sin²60° sin²80°] = [sin20° sin40° sin60° sin80°]^2. Which is the same as before. Therefore, same result.

So, if I can compute sin20° sin40° sin60° sin80°, then square it.

Earlier, I found that sin20° sin40° sin80° = √3/8, then multiplied by sin60° to get 3/16.

But let me check if that's correct. Let me compute sin20° sin40° sin60° sin80°.

Alternatively, maybe I can compute sin20° sin40° sin80° first, which we said is √3/8, then multiply by sin60°, which is √3/2.

So, √3/8 * √3/2 = (3)/16.

Alternatively, let me compute sin20° sin40° sin60° sin80° numerically. Let me calculate approximate values.

First, sin20° ≈ 0.3420, sin40° ≈ 0.6428, sin60° ≈ 0.8660, sin80° ≈ 0.9848.

Multiplying them together: 0.3420 * 0.6428 ≈ 0.2195; 0.8660 * 0.9848 ≈ 0.8536. Then multiply 0.2195 * 0.8536 ≈ 0.187.

Now, 3/16 is 0.1875. So, that's very close. Therefore, the exact value is 3/16. Therefore, squaring that gives 9/256 ≈ 0.03515625.

But let me check with another approach. Let me compute the original product numerically.

Original product: sin20° sin40° sin60° sin80° sin100° sin120° sin140° sin160°.

Compute each term:

sin20° ≈ 0.3420

sin40° ≈ 0.6428

sin60° ≈ 0.8660

sin80° ≈ 0.9848

sin100° = sin80° ≈ 0.9848

sin120° = sin60° ≈ 0.8660

sin140° = sin40° ≈ 0.6428

sin160° = sin20° ≈ 0.3420

So, multiplying all together:

First, multiply 0.3420 * 0.6428 ≈ 0.2195

Then, 0.8660 * 0.9848 ≈ 0.8536

Then, 0.9848 * 0.8660 ≈ 0.8536

Then, 0.6428 * 0.3420 ≈ 0.2195

So, the product is (0.2195 * 0.8536) * (0.8536 * 0.2195) = (approx) (0.187) * (0.187) ≈ 0.034969, which is approximately 0.035, which is 9/256 ≈ 0.03515625. So, that's very close. Therefore, the exact value is 9/256.

Therefore, the answer is 9/256.

But let me check once more if there's another way to get this result.

Alternatively, using complex numbers or roots of unity.

But maybe that's more complicated. Alternatively, using the formula for the product of sines.

I remember that the product of sin(kπ/(2n)) for k=1 to n-1 is something like sqrt(n)/2^{n-1} or similar. But here, our angles are in degrees, but maybe converting to radians.

Let me think. Let me convert all angles to radians.

20° is π/9, 40° is 2π/9, 60° is π/3, 80° is 4π/9, 100° is 5π/9, 120° is 2π/3, 140° is 7π/9, 160° is 8π/9.

So, the product is sin(π/9) sin(2π/9) sin(π/3) sin(4π/9) sin(5π/9) sin(2π/3) sin(7π/9) sin(8π/9).

Note that sin(5π/9) = sin(π - 4π/9) = sin(4π/9), similarly sin(7π/9) = sin(2π/9), sin(8π/9) = sin(π/9), and sin(2π/3) = sin(π/3). Therefore, the product becomes [sin(π/9) sin(2π/9) sin(π/3) sin(4π/9)]^2.

Which is the same as before. So, we need to compute [sin(π/9) sin(2π/9) sin(π/3) sin(4π/9)]^2.

Alternatively, maybe there's a formula for the product sin(π/9) sin(2π/9) sin(4π/9).

I recall that there is a formula for the product of sines of angles in geometric progression. For example, sinθ sin2θ sin4θ... up to n terms.

But here, we have sin(π/9) sin(2π/9) sin(4π/9). Let me check.

Let me recall that sinθ sin2θ sin4θ...sin(2^{n-1}θ) = something.

But in our case, θ = π/9, and we have sin(π/9) sin(2π/9) sin(4π/9). Which is three terms.

There is an identity that says that sinθ sin2θ sin4θ...sin(2^{n-1}θ) = (sin(2^n θ)) / 2^n.

Wait, let me check. For example, for n=1: sinθ = sin(2^1 θ)/2^1? No, that would be sinθ = sin2θ / 2, which is not true.

Wait, maybe another identity. Let me think.

Alternatively, using complex exponentials.

But maybe I can use the identity for sinθ sin(60° - θ) sin(60° + θ) again.

Wait, if I take θ = π/9, which is 20°, then sin(π/9) sin(60° - π/9) sin(60° + π/9).

Wait, 60° is π/3. So, π/3 - π/9 = 2π/9, and π/3 + π/9 = 4π/9. Therefore, sin(π/9) sin(2π/9) sin(4π/9) = sin(π/3)/4 = (√3/2)/4 = √3/8.

Which matches our previous result. Therefore, sin(π/9) sin(2π/9) sin(4π/9) = √3/8.

Then, multiplying by sin(π/3) which is √3/2, gives us √3/8 * √3/2 = 3/16.

Therefore, the product [sin(π/9) sin(2π/9) sin(π/3) sin(4π/9)] = 3/16.

Therefore, squaring that gives 9/256.

Therefore, the original product is 9/256.

Therefore, the answer is 9/256.

But let me check once more with another method.

Alternatively, using the formula for the product of sines.

There is a formula that says that the product of sin(kπ/(2n)) for k=1 to n-1 is sqrt(n)/2^{n-1} or something similar. But I need to recall the exact formula.

Alternatively, for the product of sin(kπ/(2n)) from k=1 to n-1.

But in our case, the angles are π/9, 2π/9, 4π/9, etc. Wait, but maybe not directly applicable.

Alternatively, since we have eight terms, which is 2^3, maybe there's a connection.

Alternatively, since we have already verified numerically and through identities that the product is 9/256, and the approximate calculation matches, I think that's solid.

Therefore, I think the answer is 9/256.

But just to be thorough, let me check with another identity.

Let me recall that the product of sin(kπ/n) for k=1 to n-1 is n/(2^{n-1})). Wait, but that's for k from 1 to n-1.

Wait, for example, for n=3, product sin(π/3) = √3/2, and n/(2^{n-1}) = 3/4, which is not equal. So that formula is not correct.

Wait, actually, the formula is that the product of sin(kπ/n) for k=1 to n-1 is n/(2^{n-1})). Wait, let me check for n=3: product is sin(π/3) = √3/2. n/(2^{n-1}) = 3/4. Not equal. So that formula is not correct.

Wait, maybe it's something else.

Actually, I think the correct formula is that the product of sin(kπ/n) for k=1 to n-1 is n/(2^{n-1})). Wait, but for n=2, product is sin(π/2) = 1, and n/(2^{n-1}) = 2/2 = 1. For n=3, product is sin(π/3) sin(2π/3) = (√3/2)(√3/2) = 3/4, and n/(2^{n-1}) = 3/4. So that works. For n=4, product is sin(π/4) sin(2π/4) sin(3π/4) = (√2/2)(1)(√2/2) = (2/4)(1) = 1/2. And n/(2^{n-1}) = 4/8 = 1/2. So that works.

So, the formula is that the product of sin(kπ/n) for k=1 to n-1 is n/(2^{n-1})).

But in our case, the angles are not exactly of the form kπ/n. Let me see.

Our angles are π/9, 2π/9, 4π/9, 5π/9, 7π/9, 8π/9, etc. Wait, but if we consider n=9, then the product of sin(kπ/9) for k=1 to 8 is 9/(2^8) = 9/256. Wait, that's exactly our product!

Wait, let me check.

If n=9, then the product of sin(kπ/9) for k=1 to 8 is 9/(2^8) = 9/256.

But our original product is sin(π/9) sin(2π/9) sin(4π/9) sin(5π/9) sin(7π/9) sin(8π/9) sin(π/3) sin(2π/3). Wait, no. Wait, original product is sin(π/9) sin(2π/9) sin(π/3) sin(4π/9) sin(5π/9) sin(2π/3) sin(7π/9) sin(8π/9).

But note that sin(π/3) = sin(3π/9) and sin(2π/3) = sin(6π/9). Therefore, the original product is [sin(π/9) sin(2π/9) sin(3π/9) sin(4π/9) sin(5π/9) sin(6π/9) sin(7π/9) sin(8π/9)].

Which is exactly the product of sin(kπ/9) for k=1 to 8.

Therefore, according to the formula, this product is 9/(2^8) = 9/256.

Therefore, the answer is 9/256.

Wow, that's a much more straightforward way! So, using the formula for the product of sines of the form sin(kπ/n) for k=1 to n-1, which is n/(2^{n-1}), here n=9, so 9/(2^8) = 9/256.

Therefore, the answer is 9/256.

This confirms our previous result.

Therefore, regardless of the method, the answer is 9/256.

**Final Answer**
\boxed{\dfrac{9}{256}}

To evaluate the product:

$$
\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ \sin 100^\circ \sin 120^\circ \sin 140^\circ \sin 160^\circ
$$

we begin by observing the symmetry in the angles. Each angle from $100^\circ$ to $160^\circ$ is the supplement of the corresponding angle from $20^\circ$ to $80^\circ$, i.e.,

$$
\sin 100^\circ = \sin 80^\circ, \quad \sin 120^\circ = \sin 60^\circ, \quad \sin 140^\circ = \sin 40^\circ, \quad \sin 160^\circ = \sin 20^\circ
$$

This allows us to rewrite the entire product as:

$$
(\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ)^2
$$

---

### Step 1: Evaluate the inner product

We now focus on computing:

$$
\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ
$$

We can use the identity:

$$
\sin \theta \sin (60^\circ - \theta) \sin (60^\circ + \theta) = \frac{1}{4} \sin 3\theta
$$

Let’s apply this identity with $ \theta = 20^\circ $:

$$
\sin 20^\circ \sin 40^\circ \sin 80^\circ = \frac{1}{4} \sin 60^\circ = \frac{1}{4} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{8}
$$

Now, multiply this by $ \sin 60^\circ = \frac{\sqrt{3}}{2} $:

$$
\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ = \frac{\sqrt{3}}{8} \cdot \frac{\sqrt{3}}{2} = \frac{3}{16}
$$

---

### Step 2: Square the result

Now, square the result to get the full product:

$$
(\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ)^2 = \left( \frac{3}{16} \right)^2 = \frac{9}{256}
$$

---

### Step 3: Confirm using a general identity

Alternatively, we can use a known identity for the product of sines:

$$
\prod_{k=1}^{n-1} \sin\left( \frac{k\pi}{n} \right) = \frac{n}{2^{n-1}}
$$

In our case, the angles are:

$$
\sin\left( \frac{\pi}{9} \right) \sin\left( \frac{2\pi}{9} \right) \sin\left( \frac{3\pi}{9} \right) \sin\left( \frac{4\pi}{9} \right) \sin\left( \frac{5\pi}{9} \right) \sin\left( \frac{6\pi}{9} \right) \sin\left( \frac{7\pi}{9} \right) \sin\left( \frac{8\pi}{9} \right)
$$

This is the product of $ \sin\left( \frac{k\pi}{9} \right) $ for $ k = 1 $ to $ 8 $, which corresponds to $ n = 9 $. Applying the identity:

$$
\prod_{k=1}^{8} \sin\left( \frac{k\pi}{9} \right) = \frac{9}{2^8} = \frac{9}{256}
$$

This confirms our earlier result.

---

### Final Answer

$$
\boxed{\dfrac{9}{256}}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of the product sin20° sin40° sin60° sin80° sin100° sin120° sin140° sin160°. Hmm, that's a lot of sine terms multiplied together. Let me think about how to approach this.

First, I remember that there are some trigonometric identities that can help simplify products of sines. Maybe I can pair up the terms or use some product-to-sum formulas? But with eight terms, that might get complicated. Let me see if there's a pattern or symmetry here.

Looking at the angles: 20°, 40°, 60°, 80°, 100°, 120°, 140°, 160°. These angles are all multiples of 20°, starting from 20° up to 160°, increasing by 20° each time. Wait, 20°, 40°, 60°, 80°, then 100°, which is 180° - 80°, 120° is 180° - 60°, 140° is 180° - 40°, and 160° is 180° - 20°. So, actually, the angles from 100° to 160° are supplementary angles to the first four angles.

I remember that sin(180° - x) = sinx. So, sin100° = sin80°, sin120° = sin60°, sin140° = sin40°, sin160° = sin20°. Therefore, the product can be rewritten as:

[sin20° sin40° sin60° sin80°] × [sin80° sin60° sin40° sin20°]

Which is [sin20° sin40° sin60° sin80°]^2. So, the original product is the square of the product of sin20°, sin40°, sin60°, and sin80°. That simplifies things a bit. So now I just need to compute [sin20° sin40° sin60° sin80°]^2.

But maybe there's a better way. Let me check if I can compute the product sin20° sin40° sin60° sin80° first. Let me recall some identities. There's a formula for the product of sines in an arithmetic sequence? Or maybe using complex numbers? Or perhaps using multiple-angle identities?

Alternatively, I remember that there's a formula for the product of sines of angles in arithmetic progression. But I don't remember exactly. Let me think. Alternatively, maybe I can pair terms. For example, sin20° sin80° and sin40° sin60°?

Let me try that. Let me compute sin20° sin80° first. Using the identity: sinA sinB = [cos(A - B) - cos(A + B)] / 2.

So, sin20° sin80° = [cos(20° - 80°) - cos(20° + 80°)] / 2 = [cos(-60°) - cos(100°)] / 2. Since cos(-x) = cosx, this becomes [cos60° - cos100°]/2.

Similarly, cos60° is 0.5, so that term is 0.5 - cos100°, divided by 2.

Similarly, let me compute sin40° sin60°. Using the same identity:

sin40° sin60° = [cos(40° - 60°) - cos(40° + 60°)] / 2 = [cos(-20°) - cos100°]/2 = [cos20° - cos100°]/2.

So, putting it all together, the product sin20° sin40° sin60° sin80° is equal to [ (cos60° - cos100°)/2 ] × [ (cos20° - cos100°)/2 ].

Hmm, that seems a bit messy, but maybe I can compute it. Let me write that down:

Product = [ (cos60° - cos100°)(cos20° - cos100°) ] / 4.

Let me compute the numerator:

(cos60° - cos100°)(cos20° - cos100°) = cos60° cos20° - cos60° cos100° - cos100° cos20° + cos²100°.

Hmm, this seems complicated. Maybe there's another approach. Let me think again.

Alternatively, maybe I can use the identity for sinθ sin(60° - θ) sin(60° + θ) = (sin3θ)/4. Wait, is that a valid identity? Let me check.

Yes, I think there is an identity that says sinθ sin(60° - θ) sin(60° + θ) = (sin3θ)/4. Let me verify with θ = 20°, then sin20° sin40° sin80° = sin60° / 4? Wait, sin3θ would be sin60°, which is √3/2. So, according to the identity, sin20° sin40° sin80° = (√3/2)/4 = √3/8? Let me check with actual values.

But maybe I can use this identity. Let me check if that's correct. Let me recall that formula.

Yes, the identity is: sinθ sin(60° - θ) sin(60° + θ) = (sin3θ)/4.

So, if I take θ = 20°, then sin20° sin40° sin80° = sin60° / 4.

So, sin20° sin40° sin80° = (√3/2)/4 = √3/8.

But in our case, we have sin20° sin40° sin60° sin80°. So, if I take the product sin20° sin40° sin80°, which is √3/8, and then multiply by sin60°, which is √3/2. Therefore, the product sin20° sin40° sin60° sin80° = (√3/8) × (√3/2) = (3/16).

Wait, that seems promising. Let me check again.

So, using the identity sinθ sin(60° - θ) sin(60° + θ) = sin3θ / 4.

So, if θ = 20°, then sin20° sin(60° - 20°) sin(60° + 20°) = sin20° sin40° sin80° = sin(60°)/4 = (√3/2)/4 = √3/8.

Therefore, sin20° sin40° sin80° = √3/8. Then, multiplying by sin60°, which is √3/2, gives us √3/8 × √3/2 = (3)/16.

So, sin20° sin40° sin60° sin80° = 3/16.

Therefore, the original product is [sin20° sin40° sin60° sin80°]^2 = (3/16)^2 = 9/256.

Wait, but hold on. Let me check if that's correct. Because the original product was sin20° sin40° sin60° sin80° sin100° sin120° sin140° sin160°, which we converted to [sin20° sin40° sin60° sin80°]^2.

But according to the above, that square would be (3/16)^2 = 9/256. But let me check if that's correct.

Alternatively, maybe I made a mistake in the identity. Let me verify the identity again.

Let me compute sinθ sin(60° - θ) sin(60° + θ). Let me expand it step by step.

First, compute sin(60° - θ) sin(60° + θ). Using the identity sin(A - B) sin(A + B) = sin²A - sin²B. Wait, no, that's cos(A - B) cos(A + B) = cos²A - sin²B? Wait, let me recall:

Actually, sin(A - B) sin(A + B) = sin²A - sin²B. Wait, let me check with A = 60°, B = θ. Then sin(60° - θ) sin(60° + θ) = sin²60° - sin²θ.

Yes, that's correct. Because sin(A - B) sin(A + B) = sin²A - sin²B.

Therefore, sinθ [sin²60° - sin²θ] = sinθ [ (3/4) - sin²θ ].

But according to the identity, this should equal sin3θ / 4. Let me check for θ = 20°, sin3θ = sin60° = √3/2.

So, let's compute sinθ [ (3/4) - sin²θ ] for θ = 20°.

First, sin20° ≈ 0.3420, sin²20° ≈ 0.11699. Then, 3/4 - 0.11699 ≈ 0.75 - 0.11699 ≈ 0.6330. Multiply by sin20° ≈ 0.3420 * 0.6330 ≈ 0.2166.

On the other hand, sin60° / 4 ≈ (√3/2)/4 ≈ 0.4330 / 4 ≈ 0.10825. Wait, that's not matching. Wait, so there's a discrepancy here.

Wait, so according to my calculation, sinθ [sin²60° - sin²θ] ≈ 0.2166, but according to the identity, it should be sin3θ / 4 ≈ 0.10825. So, there's a factor of 2 difference. Therefore, my previous assumption about the identity is wrong.

Hmm, so maybe the correct identity is different? Let me check again.

Alternatively, maybe I need to use another identity. Let me check the product sinθ sin(60° - θ) sin(60° + θ). Let me compute it using another method.

Let me use complex numbers or another identity. Let me recall that sin3θ = 3sinθ - 4sin³θ. But how does that relate?

Alternatively, let me compute sinθ sin(60° - θ) sin(60° + θ). Let me first compute sin(60° - θ) sin(60° + θ) as before, which is sin²60° - sin²θ. Then multiply by sinθ:

sinθ [sin²60° - sin²θ] = sinθ * (3/4 - sin²θ).

Alternatively, maybe express sinθ * sin²60° - sin³θ.

But how does that relate to sin3θ? Let me compute sin3θ.

sin3θ = 3sinθ - 4sin³θ. Therefore, 3sinθ - sin3θ = 4sin³θ.

But I have sinθ*(3/4) - sin³θ = (3/4) sinθ - sin³θ. Which is (3/4) sinθ - sin³θ.

Comparing with 3sinθ - sin3θ = 4sin³θ.

So, if I multiply both sides by 1/4: (3/4) sinθ - (1/4) sin3θ = sin³θ.

Therefore, sinθ*(3/4) - sin³θ = (1/4) sin3θ.

Therefore, sinθ [sin²60° - sin²θ] = (1/4) sin3θ.

Therefore, the identity is sinθ sin(60° - θ) sin(60° + θ) = (1/4) sin3θ.

So, earlier, I had forgotten the 1/4 factor. Therefore, for θ = 20°, sin20° sin40° sin80° = (1/4) sin60° = (1/4)(√3/2) = √3/8. Which matches the previous result.

So, my initial calculation was correct. Then, when I calculated the approximate value, I must have made a mistake. Let me check again.

For θ = 20°, sinθ [sin²60° - sin²θ] = sin20° [ ( (√3/2)^2 - sin²20° ) ] = sin20° [ 3/4 - sin²20° ] ≈ 0.3420 * [ 0.75 - 0.11699 ] ≈ 0.3420 * 0.633 ≈ 0.2166.

On the other hand, (1/4) sin60° ≈ (1/4)(0.8660) ≈ 0.2165. Which matches! So, my previous calculation was correct. So, the identity is correct.

Therefore, sin20° sin40° sin80° = √3/8. Then, multiplying by sin60°, which is √3/2, gives us √3/8 * √3/2 = (3)/16.

Therefore, sin20° sin40° sin60° sin80° = 3/16.

Therefore, the original product is [3/16]^2 = 9/256.

Wait, but let me check again. The original product was sin20° sin40° sin60° sin80° sin100° sin120° sin140° sin160°, which we converted to [sin20° sin40° sin60° sin80°]^2.

But let me confirm that. Since sin100° = sin80°, sin120° = sin60°, sin140° = sin40°, sin160° = sin20°, so the product is [sin20° sin40° sin60° sin80°] × [sin80° sin60° sin40° sin20°] = [sin20° sin40° sin60° sin80°]^2.

Yes, that's correct. Therefore, if sin20° sin40° sin60° sin80° = 3/16, then the square is 9/256.

But let me check with another approach to make sure.

Alternatively, maybe I can use the formula for the product of sines. There's a general formula for the product of sines of the form sinθ sin(60° - θ) sin(60° + θ) = (sin3θ)/4.

But in our case, we have more terms. Wait, but we already used that identity for part of the product.

Alternatively, maybe I can use the identity for multiple angles. Let me think.

Alternatively, since we have angles that are multiples of 20°, maybe we can relate them to roots of unity or something?

Alternatively, let me consider that the product of sines can be related to the formula for sin(nθ) or something.

Alternatively, let me recall that the product of sin(kπ/(2n)) for k=1 to n-1 is something... Wait, but here we have degrees, not radians. Let me convert to radians.

But maybe that's overcomplicating. Let me think again.

Alternatively, since we have eight terms, maybe we can pair them as follows:

sin20° sin160°, sin40° sin140°, sin60° sin120°, sin80° sin100°.

But since sin160° = sin20°, sin140° = sin40°, sin120° = sin60°, sin100° = sin80°, so each pair is [sin20° sin20°], [sin40° sin40°], [sin60° sin60°], [sin80° sin80°]. Wait, no. Wait, original product is sin20° sin40° sin60° sin80° sin100° sin120° sin140° sin160°, which is equal to [sin20° sin160°][sin40° sin140°][sin60° sin120°][sin80° sin100°].

Each pair is sinθ sin(180° - θ) = sinθ sinθ = sin²θ. Wait, no. Wait, sinθ sin(180° - θ) = sinθ sinθ? Wait, sin(180° - θ) = sinθ, so each pair is sinθ * sinθ = sin²θ. Therefore, the entire product is [sin²20° sin²40° sin²60° sin²80°].

Wait, but that contradicts my previous conclusion. Wait, hold on. Wait, if I pair sin20° with sin160°, which is sin20°, so sin20° * sin160° = sin²20°, similarly for others. Therefore, the entire product is [sin²20° sin²40° sin²60° sin²80°] = [sin20° sin40° sin60° sin80°]^2. Which is the same as before. Therefore, same result.

So, if I can compute sin20° sin40° sin60° sin80°, then square it.

Earlier, I found that sin20° sin40° sin80° = √3/8, then multiplied by sin60° to get 3/16.

But let me check if that's correct. Let me compute sin20° sin40° sin60° sin80°.

Alternatively, maybe I can compute sin20° sin40° sin80° first, which we said is √3/8, then multiply by sin60°, which is √3/2.

So, √3/8 * √3/2 = (3)/16.

Alternatively, let me compute sin20° sin40° sin60° sin80° numerically. Let me calculate approximate values.

First, sin20° ≈ 0.3420, sin40° ≈ 0.6428, sin60° ≈ 0.8660, sin80° ≈ 0.9848.

Multiplying them together: 0.3420 * 0.6428 ≈ 0.2195; 0.8660 * 0.9848 ≈ 0.8536. Then multiply 0.2195 * 0.8536 ≈ 0.187.

Now, 3/16 is 0.1875. So, that's very close. Therefore, the exact value is 3/16. Therefore, squaring that gives 9/256 ≈ 0.03515625.

But let me check with another approach. Let me compute the original product numerically.

Original product: sin20° sin40° sin60° sin80° sin100° sin120° sin140° sin160°.

Compute each term:

sin20° ≈ 0.3420

sin40° ≈ 0.6428

sin60° ≈ 0.8660

sin80° ≈ 0.9848

sin100° = sin80° ≈ 0.9848

sin120° = sin60° ≈ 0.8660

sin140° = sin40° ≈ 0.6428

sin160° = sin20° ≈ 0.3420

So, multiplying all together:

First, multiply 0.3420 * 0.6428 ≈ 0.2195

Then, 0.8660 * 0.9848 ≈ 0.8536

Then, 0.9848 * 0.8660 ≈ 0.8536

Then, 0.6428 * 0.3420 ≈ 0.2195

So, the product is (0.2195 * 0.8536) * (0.8536 * 0.2195) = (approx) (0.187) * (0.187) ≈ 0.034969, which is approximately 0.035, which is 9/256 ≈ 0.03515625. So, that's very close. Therefore, the exact value is 9/256.

Therefore, the answer is 9/256.

But let me check once more if there's another way to get this result.

Alternatively, using complex numbers or roots of unity.

But maybe that's more complicated. Alternatively, using the formula for the product of sines.

I remember that the product of sin(kπ/(2n)) for k=1 to n-1 is something like sqrt(n)/2^{n-1} or similar. But here, our angles are in degrees, but maybe converting to radians.

Let me think. Let me convert all angles to radians.

20° is π/9, 40° is 2π/9, 60° is π/3, 80° is 4π/9, 100° is 5π/9, 120° is 2π/3, 140° is 7π/9, 160° is 8π/9.

So, the product is sin(π/9) sin(2π/9) sin(π/3) sin(4π/9) sin(5π/9) sin(2π/3) sin(7π/9) sin(8π/9).

Note that sin(5π/9) = sin(π - 4π/9) = sin(4π/9), similarly sin(7π/9) = sin(2π/9), sin(8π/9) = sin(π/9), and sin(2π/3) = sin(π/3). Therefore, the product becomes [sin(π/9) sin(2π/9) sin(π/3) sin(4π/9)]^2.

Which is the same as before. So, we need to compute [sin(π/9) sin(2π/9) sin(π/3) sin(4π/9)]^2.

Alternatively, maybe there's a formula for the product sin(π/9) sin(2π/9) sin(4π/9).

I recall that there is a formula for the product of sines of angles in geometric progression. For example, sinθ sin2θ sin4θ... up to n terms.

But here, we have sin(π/9) sin(2π/9) sin(4π/9). Let me check.

Let me recall that sinθ sin2θ sin4θ...sin(2^{n-1}θ) = something.

But in our case, θ = π/9, and we have sin(π/9) sin(2π/9) sin(4π/9). Which is three terms.

There is an identity that says that sinθ sin2θ sin4θ...sin(2^{n-1}θ) = (sin(2^n θ)) / 2^n.

Wait, let me check. For example, for n=1: sinθ = sin(2^1 θ)/2^1? No, that would be sinθ = sin2θ / 2, which is not true.

Wait, maybe another identity. Let me think.

Alternatively, using complex exponentials.

But maybe I can use the identity for sinθ sin(60° - θ) sin(60° + θ) again.

Wait, if I take θ = π/9, which is 20°, then sin(π/9) sin(60° - π/9) sin(60° + π/9).

Wait, 60° is π/3. So, π/3 - π/9 = 2π/9, and π/3 + π/9 = 4π/9. Therefore, sin(π/9) sin(2π/9) sin(4π/9) = sin(π/3)/4 = (√3/2)/4 = √3/8.

Which matches our previous result. Therefore, sin(π/9) sin(2π/9) sin(4π/9) = √3/8.

Then, multiplying by sin(π/3) which is √3/2, gives us √3/8 * √3/2 = 3/16.

Therefore, the product [sin(π/9) sin(2π/9) sin(π/3) sin(4π/9)] = 3/16.

Therefore, squaring that gives 9/256.

Therefore, the original product is 9/256.

Therefore, the answer is 9/256.

But let me check once more with another method.

Alternatively, using the formula for the product of sines.

There is a formula that says that the product of sin(kπ/(2n)) for k=1 to n-1 is sqrt(n)/2^{n-1} or something similar. But I need to recall the exact formula.

Alternatively, for the product of sin(kπ/(2n)) from k=1 to n-1.

But in our case, the angles are π/9, 2π/9, 4π/9, etc. Wait, but maybe not directly applicable.

Alternatively, since we have eight terms, which is 2^3, maybe there's a connection.

Alternatively, since we have already verified numerically and through identities that the product is 9/256, and the approximate calculation matches, I think that's solid.

Therefore, I think the answer is 9/256.

But just to be thorough, let me check with another identity.

Let me recall that the product of sin(kπ/n) for k=1 to n-1 is n/(2^{n-1})). Wait, but that's for k from 1 to n-1.

Wait, for example, for n=3, product sin(π/3) = √3/2, and n/(2^{n-1}) = 3/4, which is not equal. So that formula is not correct.

Wait, actually, the formula is that the product of sin(kπ/n) for k=1 to n-1 is n/(2^{n-1})). Wait, let me check for n=3: product is sin(π/3) = √3/2. n/(2^{n-1}) = 3/4. Not equal. So that formula is not correct.

Wait, maybe it's something else.

Actually, I think the correct formula is that the product of sin(kπ/n) for k=1 to n-1 is n/(2^{n-1})). Wait, but for n=2, product is sin(π/2) = 1, and n/(2^{n-1}) = 2/2 = 1. For n=3, product is sin(π/3) sin(2π/3) = (√3/2)(√3/2) = 3/4, and n/(2^{n-1}) = 3/4. So that works. For n=4, product is sin(π/4) sin(2π/4) sin(3π/4) = (√2/2)(1)(√2/2) = (2/4)(1) = 1/2. And n/(2^{n-1}) = 4/8 = 1/2. So that works.

So, the formula is that the product of sin(kπ/n) for k=1 to n-1 is n/(2^{n-1})).

But in our case, the angles are not exactly of the form kπ/n. Let me see.

Our angles are π/9, 2π/9, 4π/9, 5π/9, 7π/9, 8π/9, etc. Wait, but if we consider n=9, then the product of sin(kπ/9) for k=1 to 8 is 9/(2^8) = 9/256. Wait, that's exactly our product!

Wait, let me check.

If n=9, then the product of sin(kπ/9) for k=1 to 8 is 9/(2^8) = 9/256.

But our original product is sin(π/9) sin(2π/9) sin(4π/9) sin(5π/9) sin(7π/9) sin(8π/9) sin(π/3) sin(2π/3). Wait, no. Wait, original product is sin(π/9) sin(2π/9) sin(π/3) sin(4π/9) sin(5π/9) sin(2π/3) sin(7π/9) sin(8π/9).

But note that sin(π/3) = sin(3π/9) and sin(2π/3) = sin(6π/9). Therefore, the original product is [sin(π/9) sin(2π/9) sin(3π/9) sin(4π/9) sin(5π/9) sin(6π/9) sin(7π/9) sin(8π/9)].

Which is exactly the product of sin(kπ/9) for k=1 to 8.

Therefore, according to the formula, this product is 9/(2^8) = 9/256.

Therefore, the answer is 9/256.

Wow, that's a much more straightforward way! So, using the formula for the product of sines of the form sin(kπ/n) for k=1 to n-1, which is n/(2^{n-1}), here n=9, so 9/(2^8) = 9/256.

Therefore, the answer is 9/256.

This confirms our previous result.

Therefore, regardless of the method, the answer is 9/256.

**Final Answer**
\boxed{\dfrac{9}{256}}

To evaluate the product:

$$
\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ \sin 100^\circ \sin 120^\circ \sin 140^\circ \sin 160^\circ
$$

we begin by observing the symmetry in the angles. Each angle from $100^\circ$ to $160^\circ$ is the supplement of the corresponding angle from $20^\circ$ to $80^\circ$, i.e.,

$$
\sin 100^\circ = \sin 80^\circ, \quad \sin 120^\circ = \sin 60^\circ, \quad \sin 140^\circ = \sin 40^\circ, \quad \sin 160^\circ = \sin 20^\circ
$$

This allows us to rewrite the entire product as:

$$
(\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ)^2
$$

---

### Step 1: Evaluate the inner product

We now focus on computing:

$$
\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ
$$

We can use the identity:

$$
\sin \theta \sin (60^\circ - \theta) \sin (60^\circ + \theta) = \frac{1}{4} \sin 3\theta
$$

Let’s apply this identity with $ \theta = 20^\circ $:

$$
\sin 20^\circ \sin 40^\circ \sin 80^\circ = \frac{1}{4} \sin 60^\circ = \frac{1}{4} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{8}
$$

Now, multiply this by $ \sin 60^\circ = \frac{\sqrt{3}}{2} $:

$$
\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ = \frac{\sqrt{3}}{8} \cdot \frac{\sqrt{3}}{2} = \frac{3}{16}
$$

---

### Step 2: Square the result

Now, square the result to get the full product:

$$
(\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ)^2 = \left( \frac{3}{16} \right)^2 = \frac{9}{256}
$$

---

### Step 3: Confirm using a general identity

Alternatively, we can use a known identity for the product of sines:

$$
\prod_{k=1}^{n-1} \sin\left( \frac{k\pi}{n} \right) = \frac{n}{2^{n-1}}
$$

In our case, the angles are:

$$
\sin\left( \frac{\pi}{9} \right) \sin\left( \frac{2\pi}{9} \right) \sin\left( \frac{3\pi}{9} \right) \sin\left( \frac{4\pi}{9} \right) \sin\left( \frac{5\pi}{9} \right) \sin\left( \frac{6\pi}{9} \right) \sin\left( \frac{7\pi}{9} \right) \sin\left( \frac{8\pi}{9} \right)
$$

This is the product of $ \sin\left( \frac{k\pi}{9} \right) $ for $ k = 1 $ to $ 8 $, which corresponds to $ n = 9 $. Applying the identity:

$$
\prod_{k=1}^{8} \sin\left( \frac{k\pi}{9} \right) = \frac{9}{2^8} = \frac{9}{256}
$$

This confirms our earlier result.

---

### Final Answer

$$
\boxed{\dfrac{9}{256}}
$$

Solution:
First, we know $\sin 60^\circ = \sin 120^\circ = \frac{\sqrt{3}}{2},$ so
\begin{align*}
&\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ \sin 100^\circ \sin 120^\circ \sin 140^\circ \sin 160^\circ \\
&= \frac{3}{4} \sin 20^\circ \sin 40^\circ \sin 80^\circ \sin 100^\circ \sin 140^\circ \sin 160^\circ.
\end{align*}We can then write $\sin 80^\circ = \sin 100^\circ = \cos 10^\circ,$ $\sin 140^\circ = \sin 40^\circ,$ $\sin 160^\circ = \sin 20^\circ,$ so
\begin{align*}
\frac{3}{4} \sin 20^\circ \sin 40^\circ \sin 80^\circ \sin 100^\circ \sin 140^\circ \sin 160^\circ &= \frac{3}{4} \cos^2 10^\circ \sin^2 20^\circ \sin^2 40^\circ \\
&= \frac{3}{4} (\cos 10^\circ \sin 20^\circ \sin 40^\circ)^2.
\end{align*}By product-to-sum,
\begin{align*}
\cos 10^\circ \sin 20^\circ \sin 40^\circ &= \cos 10^\circ \cdot \frac{1}{2} (\cos 20^\circ - \cos 60^\circ) \\
&= \frac{1}{2} \cos 10^\circ \left( \cos 20^\circ - \frac{1}{2} \right) \\
&= \frac{1}{2} \cos 10^\circ \cos 20^\circ - \frac{1}{4} \cos 10^\circ \\
&= \frac{1}{4} (\cos 30^\circ + \cos 10^\circ) - \frac{1}{4} \cos 10^\circ \\
&= \frac{1}{4} \cos 30^\circ \\
&= \frac{\sqrt{3}}{8}.
\end{align*}Therefore, the expression is equal to $\frac{3}{4} \left( \frac{\sqrt{3}}{8} \right)^2 = \boxed{\frac{9}{256}}.$


Explanation:
Okay, so I need to find the region of the volume enclosed by the set S, which consists of all points P such that the sum of the squares of their distances to three given planes is equal to 36. The planes are x - z = 0, x - 2y + z = 0, and x + y + z = 0.

First, I should recall the formula for the distance from a point to a plane. The distance from a point (x, y, z) to the plane ax + by + cz + d = 0 is |ax + by + cz + d| / sqrt(a² + b² + c²). But in the given planes, the equations are already set to zero, so maybe I need to adjust the formula accordingly. Let me check each plane:

1. The first plane is x - z = 0. So, in standard form, that's 1x + 0y -1z + 0 = 0. Therefore, the distance d1 from P = (x, y, z) to this plane is |x - z| / sqrt(1² + 0² + (-1)²) = |x - z| / sqrt(2).

2. The second plane is x - 2y + z = 0. So, coefficients are 1, -2, 1. Therefore, distance d2 is |x - 2y + z| / sqrt(1² + (-2)² + 1²) = |x - 2y + z| / sqrt(1 + 4 + 1) = |x - 2y + z| / sqrt(6).

3. The third plane is x + y + z = 0. Coefficients are 1, 1, 1. So, distance d3 is |x + y + z| / sqrt(1² + 1² + 1²) = |x + y + z| / sqrt(3).

So, the equation given is d1² + d2² + d3² = 36. Let me write that out:

[ (x - z)² / 2 ] + [ (x - 2y + z)² / 6 ] + [ (x + y + z)² / 3 ] = 36.

Now, I need to simplify this equation. Let me compute each term:

First term: (x - z)² / 2.

Second term: (x - 2y + z)² / 6.

Third term: (x + y + z)² / 3.

So, combining them:

Let me write them all over a common denominator. The denominators are 2, 6, 3. The least common denominator is 6. So, multiply each term accordingly:

First term: [ (x - z)² / 2 ] = 3(x - z)² / 6.

Second term: [ (x - 2y + z)² / 6 ] remains as is.

Third term: [ (x + y + z)² / 3 ] = 2(x + y + z)² / 6.

Therefore, adding them up:

[3(x - z)² + (x - 2y + z)² + 2(x + y + z)²] / 6 = 36.

Multiply both sides by 6:

3(x - z)² + (x - 2y + z)² + 2(x + y + z)² = 216.

Now, I need to expand each of these terms. Let me do that step by step.

First term: 3(x - z)².

Let me compute (x - z)² first: x² - 2xz + z². Multiply by 3: 3x² - 6xz + 3z².

Second term: (x - 2y + z)². Let me expand this. Let me denote this as (x + z - 2y)². Let me expand it as (x + z)² - 4y(x + z) + 4y². Wait, or maybe better to do it term by term:

(x - 2y + z)² = [x + z - 2y]^2 = (x + z)^2 - 4y(x + z) + 4y². Wait, but maybe expanding directly:

= x² + (-2y)^2 + z² + 2*(x*(-2y) + x*z + (-2y)*z)

Wait, no. Let me recall that (a + b + c)^2 = a² + b² + c² + 2ab + 2ac + 2bc. So here, a = x, b = -2y, c = z. Therefore:

(x - 2y + z)^2 = x² + (-2y)^2 + z² + 2*(x*(-2y) + x*z + (-2y)*z)

Compute each part:

x² + 4y² + z² + 2*(-2xy + xz - 2yz) = x² + 4y² + z² - 4xy + 2xz - 4yz.

So that's the expansion of the second term.

Third term: 2(x + y + z)^2. Let me first compute (x + y + z)^2:

= x² + y² + z² + 2xy + 2xz + 2yz. Multiply by 2:

= 2x² + 2y² + 2z² + 4xy + 4xz + 4yz.

Now, let me combine all three terms:

First term: 3x² - 6xz + 3z².

Second term: x² + 4y² + z² - 4xy + 2xz - 4yz.

Third term: 2x² + 2y² + 2z² + 4xy + 4xz + 4yz.

Now, adding them all together:

Let me collect like terms.

For x² terms:

First term: 3x²

Second term: +1x²

Third term: +2x²

Total x²: 3 + 1 + 2 = 6x².

For y² terms:

First term: 0

Second term: +4y²

Third term: +2y²

Total y²: 4 + 2 = 6y².

For z² terms:

First term: 3z²

Second term: +1z²

Third term: +2z²

Total z²: 3 + 1 + 2 = 6z².

For xy terms:

First term: 0

Second term: -4xy

Third term: +4xy

Total xy: -4xy + 4xy = 0.

For xz terms:

First term: -6xz

Second term: +2xz

Third term: +4xz

Total xz: -6 + 2 + 4 = 0xz.

For yz terms:

First term: 0

Second term: -4yz

Third term: +4yz

Total yz: -4yz + 4yz = 0.

So, all cross terms cancel out! That's interesting. So the entire expression simplifies to:

6x² + 6y² + 6z² = 216.

Divide both sides by 6:

x² + y² + z² = 36.

Wait, so the equation simplifies to the equation of a sphere with radius 6 centered at the origin?

But hold on, that seems too straightforward. Let me check my calculations again because the original problem mentions three planes, and the distances to them, but after expanding, it reduces to a sphere? That seems possible, but let me verify.

Let me check the expansion again step by step.

First term: 3(x - z)^2 = 3x² - 6xz + 3z². Correct.

Second term: (x - 2y + z)^2 = x² + 4y² + z² - 4xy + 2xz - 4yz. Let me check that again. Let me expand (x - 2y + z)^2:

Let me write it as [(x + z) - 2y]^2 = (x + z)^2 - 4y(x + z) + 4y². Then (x + z)^2 is x² + 2xz + z². Then subtract 4y(x + z) which is -4xy -4yz. Then add 4y². So altogether:

x² + 2xz + z² - 4xy - 4yz + 4y². Which is same as x² + 4y² + z² - 4xy + 2xz - 4yz. Yes, that's correct. So that term is correct.

Third term: 2(x + y + z)^2 = 2(x² + y² + z² + 2xy + 2xz + 2yz) = 2x² + 2y² + 2z² + 4xy + 4xz + 4yz. Correct.

Now adding all three terms:

First term: 3x² - 6xz + 3z²

Second term: x² + 4y² + z² - 4xy + 2xz - 4yz

Third term: 2x² + 2y² + 2z² + 4xy + 4xz + 4yz

Adding x² terms: 3x² + x² + 2x² = 6x². Correct.

y² terms: 4y² + 2y² = 6y². Correct.

z² terms: 3z² + z² + 2z² = 6z². Correct.

xy terms: -4xy + 4xy = 0. Correct.

xz terms: -6xz + 2xz + 4xz = 0. Correct.

yz terms: -4yz + 4yz = 0. Correct.

So indeed, all cross terms cancel, and we get 6x² + 6y² + 6z² = 216, which simplifies to x² + y² + z² = 36.

Therefore, the set S is the sphere centered at the origin with radius 6. Therefore, the volume enclosed by S is the volume of this sphere.

But wait, the problem says "the region of the volume enclosed by S". So, since S is the sphere, the volume enclosed by S is the volume of the sphere.

But let me think again. Is there a possibility that the equation d1² + d2² + d3² = 36 is not equivalent to x² + y² + z² = 36? Because sometimes when you square distances, there might be some absolute values or other considerations. However, in our case, when we squared the distances, we had absolute values squared, which become squares of the expressions inside. However, when we expanded, we didn't have absolute values anymore because we squared them. Wait, but in the original problem, the distances are squared, so even if the expressions inside the absolute value are negative, squaring them would make them positive. Therefore, the equation d1² + d2² + d3² = 36 is equivalent to the equation we derived, which is x² + y² + z² = 36.

But let me check with a specific point. Let me take the origin (0,0,0). What is d1, d2, d3?

For the origin:

d1 = |0 - 0| / sqrt(2) = 0.

d2 = |0 - 0 + 0| / sqrt(6) = 0.

d3 = |0 + 0 + 0| / sqrt(3) = 0.

So d1² + d2² + d3² = 0, which is not 36. So the origin is not in S. But according to our equation x² + y² + z² = 36, the origin is not on the sphere. That's okay. Let me check a point on the sphere. Let me take (6, 0, 0). Let's compute d1, d2, d3.

For (6, 0, 0):

d1 = |6 - 0| / sqrt(2) = 6 / sqrt(2).

d2 = |6 - 0 + 0| / sqrt(6) = 6 / sqrt(6).

d3 = |6 + 0 + 0| / sqrt(3) = 6 / sqrt(3).

Then d1² + d2² + d3² = (36 / 2) + (36 / 6) + (36 / 3) = 18 + 6 + 12 = 36. Which matches. So that point is on S.

Another check: take a point (0, 0, 6). Let's compute:

d1 = |0 - 6| / sqrt(2) = 6 / sqrt(2).

d2 = |0 - 0 + 6| / sqrt(6) = 6 / sqrt(6).

d3 = |0 + 0 + 6| / sqrt(3) = 6 / sqrt(3).

Same as before, so same result. So that point is also on S.

Another check: take a point (3, 0, 0). Then x² + y² + z² = 9, which is less than 36, so not on the sphere. Let's compute d1² + d2² + d3²:

d1 = |3 - 0| / sqrt(2) = 3 / sqrt(2), squared is 9/2.

d2 = |3 - 0 + 0| / sqrt(6) = 3 / sqrt(6), squared is 9/6 = 3/2.

d3 = |3 + 0 + 0| / sqrt(3) = 3 / sqrt(3) = sqrt(3), squared is 3.

Total: 9/2 + 3/2 + 3 = (9 + 3)/2 + 3 = 6 + 3 = 9, which is 9, which is equal to x² + y² + z². So indeed, the equation d1² + d2² + d3² equals x² + y² + z². Therefore, the equation d1² + d2² + d3² = 36 is equivalent to x² + y² + z² = 36.

Therefore, the set S is the sphere of radius 6 centered at the origin. Therefore, the volume enclosed by S is the volume of this sphere.

The volume of a sphere is (4/3)πr³. Here, r = 6, so volume is (4/3)π*(6)^3 = (4/3)π*216 = 288π.

But wait, the problem says "the region of the volume enclosed by S". Since S is the sphere, the volume enclosed by S is the volume inside the sphere. Therefore, the answer should be 288π.

But let me check again if there's any mistake here. The key step was realizing that the sum of squares of distances to those three planes simplifies to the equation of a sphere. But why does that happen? Is there a geometric interpretation?

Alternatively, maybe there's a coordinate transformation or something. But since the algebra checks out with specific points, and the expansion leads to x² + y² + z² = 36, I think that's correct.

Alternatively, maybe the problem is designed to show that the sum of squares of distances to these planes is equivalent to the square of the distance to the origin? Wait, but in general, the sum of squares of distances to planes isn't directly related to the distance to the origin. However, in this case, due to the specific planes, it simplifies.

Alternatively, maybe the planes are related to coordinate planes or something else? Let me check the planes again:

First plane: x - z = 0, which is the plane where x = z. Second plane: x - 2y + z = 0. Third plane: x + y + z = 0.

But regardless of their orientation, the algebra shows that the equation simplifies to a sphere. Therefore, unless there's a miscalculation in expanding, which I checked with specific points, the conclusion is correct.

Therefore, the volume enclosed by S is the volume of the sphere with radius 6, which is (4/3)π*(6)^3 = 288π.

But let me check once more the expansion. Let me recompute the entire expression:

Original equation after expanding:

6x² + 6y² + 6z² = 216 => x² + y² + z² = 36.

Yes. Therefore, the answer is 288π.

But the problem says "Find the region of the volume enclosed by S." So, since S is the sphere, the region is the interior of the sphere. Therefore, the volume is 288π.

But let me check if the problem is in three dimensions. Yes, it's in three dimensions. So, the answer is 288π.

But wait, the problem says "the region of the volume enclosed by S". So, maybe they want the answer in terms of a sphere? But the question says "Find the region of the volume enclosed by S." So, since S is the sphere, the volume enclosed is the sphere itself. But the problem says "the region of the volume enclosed by S", so maybe they just want the volume, which is 288π.

But let me check if there's a possibility that S is not the entire sphere. For example, maybe the equation d1² + d2² + d3² = 36 is equivalent to x² + y² + z² = 36, but maybe there are some constraints? For example, if the distances are squared, but the original distances are absolute values. However, when we square them, the absolute value is squared, so it's equivalent to the square of the expression without absolute value. Therefore, the equation is equivalent.

Therefore, I think the answer is 288π.

But let me check once more with another point. Let me take a point on the sphere, say (0, 0, 6). As before, the distances squared sum to 36. Let me take another point, say (sqrt(36), 0, 0) = (6, 0, 0). As before, same result.

Alternatively, take a point (3, 4, 0). Then x² + y² + z² = 9 + 16 + 0 = 25, which is less than 36. Let me compute d1² + d2² + d3²:

d1 = |3 - 0| / sqrt(2) = 3 / sqrt(2), squared is 9/2.

d2 = |3 - 0 + 0| / sqrt(6) = 3 / sqrt(6), squared is 9/6 = 3/2.

d3 = |3 + 4 + 0| / sqrt(3) = 7 / sqrt(3), squared is 49/3.

Total: 9/2 + 3/2 + 49/3 = (12/2) + 49/3 = 6 + 49/3 = (18 + 49)/3 = 67/3 ≈ 22.33, which is not 36. However, x² + y² + z² = 25, which is not 36. So, that's consistent.

Another point: take (sqrt(36), 0, 0) = (6, 0, 0). As before, sum is 36.

Therefore, the equation is correct. Therefore, the answer is the volume of the sphere with radius 6, which is 288π.

But let me check the formula again. Volume of sphere is (4/3)πr³. For r = 6, that is (4/3)π*(6)^3 = (4/3)*π*216 = 4*72*π = 288π. Yes.

Therefore, the answer is 288π.

But the problem says "Find the region of the volume enclosed by S." So, maybe they want the answer as a sphere? But the question says "the region of the volume enclosed by S", so the volume is 288π.

But let me check if there's any trick here. For example, maybe the set S is not the entire sphere? For example, maybe the equation d1² + d2² + d3² = 36 is equivalent to x² + y² + z² = 36, but maybe there are some restrictions? For example, if the planes are not all intersecting at the origin? Wait, the planes x - z = 0, x - 2y + z = 0, and x + y + z = 0. Let me check if they intersect at the origin.

For the origin (0,0,0), plugging into all three equations: 0 - 0 = 0, 0 - 0 + 0 = 0, 0 + 0 + 0 = 0. So yes, they all pass through the origin. Therefore, the origin is a common point.

But even so, the distances from any point to the planes are non-negative, but when we square them, they are positive. Therefore, the equation is equivalent to x² + y² + z² = 36.

Therefore, I think the answer is 288π.

But let me check once more if the initial equation was correctly converted.

Original equation:

d1² + d2² + d3² = 36.

Which became:

[(x - z)^2]/2 + [(x - 2y + z)^2]/6 + [(x + y + z)^2]/3 = 36.

Then multiplied by 6:

3(x - z)^2 + (x - 2y + z)^2 + 2(x + y + z)^2 = 216.

Then expanding gives 6x² + 6y² + 6z² = 216 => x² + y² + z² = 36.

Yes.

Therefore, the answer is the volume of the sphere with radius 6, which is 288π.

Therefore, the region enclosed by S is a sphere with radius 6, and the volume is 288π.

**Final Answer**
The volume enclosed by $ S $ is \boxed{288\pi}.

To determine the region of the volume enclosed by the set $ S $, we begin by analyzing the given condition:

$$
d_1^2 + d_2^2 + d_3^2 = 36
$$

where $ d_1, d_2, d_3 $ are the distances from a point $ P = (x, y, z) $ to the planes:

- $ x - z = 0 $
- $ x - 2y + z = 0 $
- $ x + y + z = 0 $

---

### Step 1: Express the Distances

Using the standard formula for the distance from a point to a plane $ ax + by + cz + d = 0 $, we compute:

- $ d_1 = \frac{|x - z|}{\sqrt{2}} $
- $ d_2 = \frac{|x - 2y + z|}{\sqrt{6}} $
- $ d_3 = \frac{|x + y + z|}{\sqrt{3}} $

Squaring each distance:

$$
d_1^2 = \frac{(x - z)^2}{2}, \quad d_2^2 = \frac{(x - 2y + z)^2}{6}, \quad d_3^2 = \frac{(x + y + z)^2}{3}
$$

Substituting into the given equation:

$$
\frac{(x - z)^2}{2} + \frac{(x - 2y + z)^2}{6} + \frac{(x + y + z)^2}{3} = 36
$$

---

### Step 2: Combine the Terms

To simplify, we find a common denominator of 6:

$$
\frac{3(x - z)^2 + (x - 2y + z)^2 + 2(x + y + z)^2}{6} = 36
$$

Multiply both sides by 6:

$$
3(x - z)^2 + (x - 2y + z)^2 + 2(x + y + z)^2 = 216
$$

Now expand each term:

- $ (x - z)^2 = x^2 - 2xz + z^2 $
- $ (x - 2y + z)^2 = x^2 + 4y^2 + z^2 - 4xy + 2xz - 4yz $
- $ (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz $

Multiply by the respective coefficients:

- $ 3(x - z)^2 = 3x^2 - 6xz + 3z^2 $
- $ (x - 2y + z)^2 = x^2 + 4y^2 + z^2 - 4xy + 2xz - 4yz $
- $ 2(x + y + z)^2 = 2x^2 + 2y^2 + 2z^2 + 4xy + 4xz + 4yz $

Now sum all terms:

- $ x^2 $: $ 3x^2 + x^2 + 2x^2 = 6x^2 $
- $ y^2 $: $ 4y^2 + 2y^2 = 6y^2 $
- $ z^2 $: $ 3z^2 + z^2 + 2z^2 = 6z^2 $
- $ xy $: $ -4xy + 4xy = 0 $
- $ xz $: $ -6xz + 2xz + 4xz = 0 $
- $ yz $: $ -4yz + 4yz = 0 $

So the entire expression simplifies to:

$$
6x^2 + 6y^2 + 6z^2 = 216
$$

Divide both sides by 6:

$$
x^2 + y^2 + z^2 = 36
$$

---

### Step 3: Interpret the Result

This is the equation of a **sphere** centered at the origin with radius $ r = \sqrt{36} = 6 $.

Thus, the set $ S $ is the **surface** of this sphere, and the **region of the volume enclosed by $ S $** is the **interior** of the sphere.

---

### Step 4: Compute the Volume

The volume $ V $ of a sphere of radius $ r $ is given by:

$$
V = \frac{4}{3} \pi r^3
$$

Substitute $ r = 6 $:

$$
V = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288\pi
$$

---

### Final Answer

$$
\boxed{288\pi}
$$

[Assistant]
Explanation:
Okay, so I need to find the region of the volume enclosed by the set S, which consists of all points P such that the sum of the squares of their distances to three given planes is equal to 36. The planes are x - z = 0, x - 2y + z = 0, and x + y + z = 0.

First, I should recall the formula for the distance from a point to a plane. The distance from a point (x, y, z) to the plane ax + by + cz + d = 0 is |ax + by + cz + d| / sqrt(a² + b² + c²). But in the given planes, the equations are already set to zero, so maybe I need to adjust the formula accordingly. Let me check each plane:

1. The first plane is x - z = 0. So, in standard form, that's 1x + 0y -1z + 0 = 0. Therefore, the distance d1 from P = (x, y, z) to this plane is |x - z| / sqrt(1² + 0² + (-1)²) = |x - z| / sqrt(2).

2. The second plane is x - 2y + z = 0. So, coefficients are 1, -2, 1. Therefore, distance d2 is |x - 2y + z| / sqrt(1² + (-2)² + 1²) = |x - 2y + z| / sqrt(1 + 4 + 1) = |x - 2y + z| / sqrt(6).

3. The third plane is x + y + z = 0. Coefficients are 1, 1, 1. So, distance d3 is |x + y + z| / sqrt(1² + 1² + 1²) = |x + y + z| / sqrt(3).

So, the equation given is d1² + d2² + d3² = 36. Let me write that out:

[ (x - z)² / 2 ] + [ (x - 2y + z)² / 6 ] + [ (x + y + z)² / 3 ] = 36.

Now, I need to simplify this equation. Let me compute each term:

First term: (x - z)² / 2.

Second term: (x - 2y + z)² / 6.

Third term: (x + y + z)² / 3.

So, combining them:

Let me write them all over a common denominator. The denominators are 2, 6, 3. The least common denominator is 6. So, multiply each term accordingly:

First term: [ (x - z)² / 2 ] = 3(x - z)² / 6.

Second term: [ (x - 2y + z)² / 6 ] remains as is.

Third term: [ (x + y + z)² / 3 ] = 2(x + y + z)² / 6.

Therefore, adding them up:

[3(x - z)² + (x - 2y + z)² + 2(x + y + z)²] / 6 = 36.

Multiply both sides by 6:

3(x - z)² + (x - 2y + z)² + 2(x + y + z)² = 216.

Now, I need to expand each of these terms. Let me do that step by step.

First term: 3(x - z)².

Let me compute (x - z)² first: x² - 2xz + z². Multiply by 3: 3x² - 6xz + 3z².

Second term: (x - 2y + z)². Let me expand this. Let me denote this as (x + z - 2y)². Let me expand it as (x + z)² - 4y(x + z) + 4y². Wait, or maybe better to do it term by term:

(x - 2y + z)² = [x + z - 2y]^2 = (x + z)^2 - 4y(x + z) + 4y². Wait, but maybe expanding directly:

= x² + (-2y)^2 + z² + 2*(x*(-2y) + x*z + (-2y)*z)

Wait, no. Let me recall that (a + b + c)^2 = a² + b² + c² + 2ab + 2ac + 2bc. So here, a = x, b = -2y, c = z. Therefore:

(x - 2y + z)^2 = x² + (-2y)^2 + z² + 2*(x*(-2y) + x*z + (-2y)*z)

Compute each part:

x² + 4y² + z² + 2*(-2xy + xz - 2yz) = x² + 4y² + z² - 4xy + 2xz - 4yz.

So that's the expansion of the second term.

Third term: 2(x + y + z)^2. Let me first compute (x + y + z)^2:

= x² + y² + z² + 2xy + 2xz + 2yz. Multiply by 2:

= 2x² + 2y² + 2z² + 4xy + 4xz + 4yz.

Now, let me combine all three terms:

First term: 3x² - 6xz + 3z².

Second term: x² + 4y² + z² - 4xy + 2xz - 4yz.

Third term: 2x² + 2y² + 2z² + 4xy + 4xz + 4yz.

Now, adding them all together:

Let me collect like terms.

For x² terms:

First term: 3x²

Second term: +1x²

Third term: +2x²

Total x²: 3 + 1 + 2 = 6x².

For y² terms:

First term: 0

Second term: +4y²

Third term: +2y²

Total y²: 4 + 2 = 6y².

For z² terms:

First term: 3z²

Second term: +1z²

Third term: +2z²

Total z²: 3 + 1 + 2 = 6z².

For xy terms:

First term: 0

Second term: -4xy

Third term: +4xy

Total xy: -4xy + 4xy = 0.

For xz terms:

First term: -6xz

Second term: +2xz

Third term: +4xz

Total xz: -6 + 2 + 4 = 0xz.

For yz terms:

First term: 0

Second term: -4yz

Third term: +4yz

Total yz: -4yz + 4yz = 0.

So, all cross terms cancel out! That's interesting. So the entire expression simplifies to:

6x² + 6y² + 6z² = 216.

Divide both sides by 6:

x² + y² + z² = 36.

Wait, so the equation simplifies to the equation of a sphere with radius 6 centered at the origin?

But hold on, that seems too straightforward. Let me check my calculations again because the original problem mentions three planes, and the distances to them, but after expanding, it reduces to a sphere? That seems possible, but let me verify.

Let me check the expansion again step by step.

First term: 3(x - z)^2 = 3x² - 6xz + 3z². Correct.

Second term: (x - 2y + z)^2 = x² + 4y² + z² - 4xy + 2xz - 4yz. Let me check that again. Let me expand (x - 2y + z)^2:

Let me write it as [(x + z) - 2y]^2 = (x + z)^2 - 4y(x + z) + 4y². Then (x + z)^2 is x² + 2xz + z². Then subtract 4y(x + z) which is -4xy -4yz. Then add 4y². So altogether:

x² + 2xz + z² - 4xy - 4yz + 4y². Which is same as x² + 4y² + z² - 4xy + 2xz - 4yz. Yes, that's correct. So that term is correct.

Third term: 2(x + y + z)^2 = 2(x² + y² + z² + 2xy + 2xz + 2yz) = 2x² + 2y² + 2z² + 4xy + 4xz + 4yz. Correct.

Now adding all three terms:

First term: 3x² - 6xz + 3z²

Second term: x² + 4y² + z² - 4xy + 2xz - 4yz

Third term: 2x² + 2y² + 2z² + 4xy + 4xz + 4yz

Adding x² terms: 3x² + x² + 2x² = 6x². Correct.

y² terms: 4y² + 2y² = 6y². Correct.

z² terms: 3z² + z² + 2z² = 6z². Correct.

xy terms: -4xy + 4xy = 0. Correct.

xz terms: -6xz + 2xz + 4xz = 0. Correct.

yz terms: -4yz + 4yz = 0. Correct.

So indeed, all cross terms cancel, and we get 6x² + 6y² + 6z² = 216, which simplifies to x² + y² + z² = 36.

Therefore, the set S is the sphere centered at the origin with radius 6. Therefore, the volume enclosed by S is the volume of this sphere.

But wait, the problem says "the region of the volume enclosed by S". So, since S is the sphere, the volume enclosed by S is the volume of the sphere.

But let me think again. Is there a possibility that the equation d1² + d2² + d3² = 36 is not equivalent to x² + y² + z² = 36? Because sometimes when you square distances, there might be some absolute values or other considerations. However, in our case, when we squared the distances, we had absolute values squared, which become squares of the expressions inside. However, when we expanded, we didn't have absolute values anymore because we squared them. Wait, but in the original problem, the distances are squared, so even if the expressions inside the absolute value are negative, squaring them would make them positive. Therefore, the equation d1² + d2² + d3² = 36 is equivalent to the equation we derived, which is x² + y² + z² = 36.

But let me check with a specific point. Let me take the origin (0,0,0). What is d1, d2, d3?

For the origin:

d1 = |0 - 0| / sqrt(2) = 0.

d2 = |0 - 0 + 0| / sqrt(6) = 0.

d3 = |0 + 0 + 0| / sqrt(3) = 0.

So d1² + d2² + d3² = 0, which is not 36. So the origin is not in S. But according to our equation x² + y² + z² = 36, the origin is not on the sphere. That's okay. Let me check a point on the sphere. Let me take (6, 0, 0). Let's compute d1, d2, d3.

For (6, 0, 0):

d1 = |6 - 0| / sqrt(2) = 6 / sqrt(2).

d2 = |6 - 0 + 0| / sqrt(6) = 6 / sqrt(6).

d3 = |6 + 0 + 0| / sqrt(3) = 6 / sqrt(3).

Then d1² + d2² + d3² = (36 / 2) + (36 / 6) + (36 / 3) = 18 + 6 + 12 = 36. Which matches. So that point is on S.

Another check: take a point (0, 0, 6). Let's compute:

d1 = |0 - 6| / sqrt(2) = 6 / sqrt(2).

d2 = |0 - 0 + 6| / sqrt(6) = 6 / sqrt(6).

d3 = |0 + 0 + 6| / sqrt(3) = 6 / sqrt(3).

Same as before, so same result. So that point is also on S.

Another check: take a point (3, 0, 0). Then x² + y² + z² = 9, which is less than 36, so not on the sphere. Let's compute d1² + d2² + d3²:

d1 = |3 - 0| / sqrt(2) = 3 / sqrt(2), squared is 9/2.

d2 = |3 - 0 + 0| / sqrt(6) = 3 / sqrt(6), squared is 9/6 = 3/2.

d3 = |3 + 0 + 0| / sqrt(3) = 3 / sqrt(3) = sqrt(3), squared is 3.

Total: 9/2 + 3/2 + 3 = (9 + 3)/2 + 3 = 6 + 3 = 9, which is 9, which is equal to x² + y² + z². So indeed, the equation d1² + d2² + d3² equals x² + y² + z². Therefore, the equation d1² + d2² + d3² = 36 is equivalent to x² + y² + z² = 36.

Therefore, the set S is the sphere of radius 6 centered at the origin. Therefore, the volume enclosed by S is the volume of this sphere.

The volume of a sphere is (4/3)πr³. Here, r = 6, so volume is (4/3)π*(6)^3 = (4/3)π*216 = 288π.

But wait, the problem says "the region of the volume enclosed by S". Since S is the sphere, the volume enclosed by S is the volume inside the sphere. Therefore, the answer should be 288π.

But let me check again if there's any mistake here. The key step was realizing that the sum of squares of distances to those three planes simplifies to the equation of a sphere. But why does that happen? Is there a geometric interpretation?

Alternatively, maybe there's a coordinate transformation or something. But since the algebra checks out with specific points, and the expansion leads to x² + y² + z² = 36, I think that's correct.

Alternatively, maybe the problem is designed to show that the sum of squares of distances to these planes is equivalent to the square of the distance to the origin? Wait, but in general, the sum of squares of distances to planes isn't directly related to the distance to the origin. However, in this case, due to the specific planes, it simplifies.

Alternatively, maybe the planes are related to coordinate planes or something else? Let me check the planes again:

First plane: x - z = 0, which is the plane where x = z. Second plane: x - 2y + z = 0. Third plane: x + y + z = 0.

But regardless of their orientation, the algebra shows that the equation simplifies to a sphere. Therefore, unless there's a miscalculation in expanding, which I checked with specific points, the conclusion is correct.

Therefore, the volume enclosed by S is the volume of the sphere with radius 6, which is (4/3)π*(6)^3 = 288π.

But let me check once more the expansion. Let me recompute the entire expression:

Original equation after expanding:

6x² + 6y² + 6z² = 216 => x² + y² + z² = 36.

Yes. Therefore, the answer is 288π.

But the problem says "Find the region of the volume enclosed by S." So, since S is the sphere, the region is the interior of the sphere. Therefore, the volume is 288π.

But let me check if the problem is in three dimensions. Yes, it's in three dimensions. So, the answer is 288π.

But wait, the problem says "the region of the volume enclosed by S". So, maybe they want the answer in terms of a sphere? But the question says "Find the region of the volume enclosed by S." So, since S is the sphere, the volume enclosed is the sphere itself. But the problem says "the region of the volume enclosed by S", so maybe they just want the volume, which is 288π.

But let me check if there's a possibility that S is not the entire sphere. For example, maybe the equation d1² + d2² + d3² = 36 is equivalent to x² + y² + z² = 36, but maybe there are some constraints? For example, if the distances are squared, but the original distances are absolute values. However, when we square them, the absolute value is squared, so it's equivalent to the square of the expression without absolute value. Therefore, the equation is equivalent.

Therefore, I think the answer is 288π.

But let me check once more with another point. Let me take a point on the sphere, say (0, 0, 6). As before, the distances squared sum to 36. Let me take another point, say (sqrt(36), 0, 0) = (6, 0, 0). As before, same result.

Alternatively, take a point (3, 4, 0). Then x² + y² + z² = 9 + 16 + 0 = 25, which is less than 36. Let me compute d1² + d2² + d3²:

d1 = |3 - 0| / sqrt(2) = 3 / sqrt(2), squared is 9/2.

d2 = |3 - 0 + 0| / sqrt(6) = 3 / sqrt(6), squared is 9/6 = 3/2.

d3 = |3 + 4 + 0| / sqrt(3) = 7 / sqrt(3), squared is 49/3.

Total: 9/2 + 3/2 + 49/3 = (12/2) + 49/3 = 6 + 49/3 = (18 + 49)/3 = 67/3 ≈ 22.33, which is not 36. However, x² + y² + z² = 25, which is not 36. So, that's consistent.

Another point: take (sqrt(36), 0, 0) = (6, 0, 0). As before, sum is 36.

Therefore, the equation is correct. Therefore, the answer is the volume of the sphere with radius 6, which is 288π.

But let me check the formula again. Volume of sphere is (4/3)πr³. For r = 6, that is (4/3)π*(6)^3 = (4/3)*π*216 = 4*72*π = 288π. Yes.

Therefore, the answer is 288π.

But the problem says "Find the region of the volume enclosed by S." So, maybe they want the answer as a sphere? But the question says "the region of the volume enclosed by S", so the volume is 288π.

But let me check if there's any trick here. For example, maybe the set S is not the entire sphere? For example, maybe the equation d1² + d2² + d3² = 36 is equivalent to x² + y² + z² = 36, but maybe there are some restrictions? For example, if the planes are not all intersecting at the origin? Wait, the planes x - z = 0, x - 2y + z = 0, and x + y + z = 0. Let me check if they intersect at the origin.

For the origin (0,0,0), plugging into all three equations: 0 - 0 = 0, 0 - 0 + 0 = 0, 0 + 0 + 0 = 0. So yes, they all pass through the origin. Therefore, the origin is a common point.

But even so, the distances from any point to the planes are non-negative, but when we square them, they are positive. Therefore, the equation is equivalent to x² + y² + z² = 36.

Therefore, I think the answer is 288π.

But let me check once more if the initial equation was correctly converted.

Original equation:

d1² + d2² + d3² = 36.

Which became:

[(x - z)^2]/2 + [(x - 2y + z)^2]/6 + [(x + y + z)^2]/3 = 36.

Then multiplied by 6:

3(x - z)^2 + (x - 2y + z)^2 + 2(x + y + z)^2 = 216.

Then expanding gives 6x² + 6y² + 6z² = 216 => x² + y² + z² = 36.

Yes.

Therefore, the answer is the volume of the sphere with radius 6, which is 288π.

Therefore, the region enclosed by S is a sphere with radius 6, and the volume is 288π.

**Final Answer**
The volume enclosed by $ S $ is \boxed{288\pi}.

To determine the region of the volume enclosed by the set $ S $, we begin by analyzing the given condition:

$$
d_1^2 + d_2^2 + d_3^2 = 36
$$

where $ d_1, d_2, d_3 $ are the distances from a point $ P = (x, y, z) $ to the planes:

- $ x - z = 0 $
- $ x - 2y + z = 0 $
- $ x + y + z = 0 $

---

### Step 1: Express the Distances

Using the standard formula for the distance from a point to a plane $ ax + by + cz + d = 0 $, we compute:

- $ d_1 = \frac{|x - z|}{\sqrt{2}} $
- $ d_2 = \frac{|x - 2y + z|}{\sqrt{6}} $
- $ d_3 = \frac{|x + y + z|}{\sqrt{3}} $

Squaring each distance:

$$
d_1^2 = \frac{(x - z)^2}{2}, \quad d_2^2 = \frac{(x - 2y + z)^2}{6}, \quad d_3^2 = \frac{(x + y + z)^2}{3}
$$

Substituting into the given equation:

$$
\frac{(x - z)^2}{2} + \frac{(x - 2y + z)^2}{6} + \frac{(x + y + z)^2}{3} = 36
$$

---

### Step 2: Combine the Terms

To simplify, we find a common denominator of 6:

$$
\frac{3(x - z)^2 + (x - 2y + z)^2 + 2(x + y + z)^2}{6} = 36
$$

Multiply both sides by 6:

$$
3(x - z)^2 + (x - 2y + z)^2 + 2(x + y + z)^2 = 216
$$

Now expand each term:

- $ (x - z)^2 = x^2 - 2xz + z^2 $
- $ (x - 2y + z)^2 = x^2 + 4y^2 + z^2 - 4xy + 2xz - 4yz $
- $ (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz $

Multiply by the respective coefficients:

- $ 3(x - z)^2 = 3x^2 - 6xz + 3z^2 $
- $ (x - 2y + z)^2 = x^2 + 4y^2 + z^2 - 4xy + 2xz - 4yz $
- $ 2(x + y + z)^2 = 2x^2 + 2y^2 + 2z^2 + 4xy + 4xz + 4yz $

Now sum all terms:

- $ x^2 $: $ 3x^2 + x^2 + 2x^2 = 6x^2 $
- $ y^2 $: $ 4y^2 + 2y^2 = 6y^2 $
- $ z^2 $: $ 3z^2 + z^2 + 2z^2 = 6z^2 $
- $ xy $: $ -4xy + 4xy = 0 $
- $ xz $: $ -6xz + 2xz + 4xz = 0 $
- $ yz $: $ -4yz + 4yz = 0 $

So the entire expression simplifies to:

$$
6x^2 + 6y^2 + 6z^2 = 216
$$

Divide both sides by 6:

$$
x^2 + y^2 + z^2 = 36
$$

---

### Step 3: Interpret the Result

This is the equation of a **sphere** centered at the origin with radius $ r = \sqrt{36} = 6 $.

Thus, the set $ S $ is the **surface** of this sphere, and the **region of the volume enclosed by $ S $** is the **interior** of the sphere.

---

### Step 4: Compute the Volume

The volume $ V $ of a sphere of radius $ r $ is given by:

$$
V = \frac{4}{3} \pi r^3
$$

Substitute $ r = 6 $:

$$
V = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288\pi
$$

---

### Final Answer

$$
\boxed{288\pi}
$$

Solution:
Let $P = (a,b,c).$  Then the distance from $P$ to the plane $x - z = 0$ is
\[d_1 = \frac{|a - c|}{\sqrt{1^2 + (-1)^2}} = \frac{|a - c|}{\sqrt{2}}.\]The distance from $P$ to the plane $x - 2y + z = 0$ is
\[d_2 = \frac{|a - 2b + c|}{\sqrt{1^2 + (-2)^2 + 1^2}} = \frac{|a - 2b + c|}{\sqrt{6}}.\]And, the distance from $P$ to the plane $x + y + z = 0$ is
\[d_3 = \frac{|a + b + c|}{\sqrt{1^2 + 1^2 + 1^2}} = \frac{|a + b + c|}{\sqrt{3}}.\]Then the equation $d_1^2 + d_2^2 + d_3^2 = 36$ becomes
\[\frac{(a - c)^2}{2} + \frac{(a - 2b + c)^2}{6} + \frac{(a + b + c)^2}{3} = 36.\]This simplifies to $a^2 + b^2 + c^2 = 36.$  Thus, $S$ is a sphere with radius 6, so its volume is
\[\frac{4}{3} \pi \cdot 6^3 = \boxed{288 \pi}.\]


Explanation:
Okay, so Yann and Camille are going to a restaurant, and there are 10 items on the menu. Each of them orders one dish. The question is asking how many different combinations of meals they can order if they refuse to order the same dish. Also, it mentions that it does matter who orders what, so Yann ordering chicken and Camille ordering fish is different from Yann ordering fish and Camille ordering chicken. Alright, let me think through this step by step.

First, let me make sure I understand the problem correctly. There are 10 dishes on the menu. Yann and Camille each choose one dish. They can't choose the same dish. And since it matters who orders what, the order of selection is important. So, for example, if Yann picks dish A and Camille picks dish B, that's a different combination than Yann picking dish B and Camille picking dish A. So, we need to count all such possible ordered pairs where the two dishes are different.

Let me recall some combinatorial principles here. If there were no restrictions, how many combinations would there be? Well, since each person can choose any dish, and their choices are independent, it would be 10 choices for Yann and 10 choices for Camille, so 10*10 = 100 total combinations. But since they can't order the same dish, we need to subtract the cases where they order the same dish.

Alternatively, maybe it's easier to think of it as permutations. Since the order matters, and they can't have the same dish. Wait, but permutations usually involve arranging items where order matters, but here we have two people choosing dishes. Let me think.

Another approach: For Yann, he can choose any of the 10 dishes. Once Yann has chosen a dish, Camille cannot choose that same dish. So, Camille has 9 remaining dishes to choose from. Therefore, the total number of combinations would be 10 * 9 = 90. Is that right?

Wait, let me check. If Yann picks first, there are 10 options. For each of Yann's choices, Camille has 9 options. So, 10*9=90. Alternatively, if we think of it as permutations of 10 dishes taken 2 at a time, which is 10*9=90. Since permutations account for order, which in this case is important because Yann and Camille are distinct individuals. So, that seems to fit.

But let me verify with another approach. Let's think about all possible ordered pairs (Yann's dish, Camille's dish) where the two dishes are different. The total number of ordered pairs is 10*10=100. The number of ordered pairs where they have the same dish is 10 (one for each dish: both choose dish 1, both choose dish 2, etc.). Therefore, subtracting those, we get 100 - 10 = 90. Yep, same answer. So that seems to confirm it.

Alternatively, if I think about combinations versus permutations. If the problem had said that the order doesn't matter, like how many ways can they order two different dishes regardless of who gets which, then it would be combinations. In that case, it would be C(10,2) = 45. But since the problem says that it does matter who orders what, so each of those 45 combinations would correspond to two different ordered pairs. For example, choosing dish A and dish B can be arranged in two ways: Yann has A and Camille has B, or vice versa. Therefore, 45*2=90. Which again gives the same answer. So that's another way to think about it.

So, all three approaches—direct counting, subtracting the same dish cases, and thinking in terms of combinations and then accounting for order—lead me to 90. Therefore, I think the answer is 90.

But let me just make sure there isn't a trick or something I'm missing. The problem says "different combinations of meals". Wait, does "combinations" here mean something different? Wait, in common language, sometimes "combinations" can be ambiguous. But in combinatorics, "combinations" usually refer to unordered selections, while "permutations" are ordered. However, the problem specifically says "It does matter who orders what---Yann ordering chicken and Camille ordering fish is different from Yann ordering fish and Camille ordering chicken." So, they are emphasizing that the order matters. Therefore, the problem is not asking for combinations in the combinatorial sense, but rather permutations. So, the answer is permutations of 10 dishes taken 2 at a time, which is 10*9=90. So, that's consistent.

Alternatively, if they had said "how many different combinations of meals can they order", without specifying that it matters who orders what, then it might be combinations. But since they specifically mention that it does matter, we have to consider the order. Therefore, 90 is correct.

Another way to think about it: For each of the 10 dishes Yann can choose, Camille has 9 options. So, 10*9=90. So, that's straightforward.

Alternatively, if we think of the first person (Yann) has 10 choices, the second person (Camille) has 9 choices, so 10*9=90. So, that's the same as before.

Alternatively, if we think of the problem as assigning dishes to Yann and Camille, with the constraint that they are different. So, the number of injective functions from the set {Yann, Camille} to the set of 10 dishes. The number of injective functions is P(10,2)=10*9=90. So, that's another way to think about it.

Therefore, all these different ways of thinking about it lead me to the same answer. Therefore, I feel confident that the answer is 90.

But just to make sure, let me take a smaller example. Suppose there are 2 dishes on the menu, say A and B. How many combinations? According to the formula, it would be 2*1=2. Let's check. The possibilities are:

Yann orders A, Camille orders B.

Yann orders B, Camille orders A.

That's two combinations, which is correct. If we didn't have the restriction, it would be 2*2=4. Subtract the cases where they order the same dish: 2 cases (both A or both B). So 4-2=2. Correct. So, that works.

Another example: 3 dishes. Then according to formula, 3*2=6. Let's list them:

Yann A, Camille B

Yann A, Camille C

Yann B, Camille A

Yann B, Camille C

Yann C, Camille A

Yann C, Camille B

That's 6, which is correct. If we didn't have the restriction, it would be 9. Subtract 3 cases where they order the same dish, so 6. Correct. So, that seems to check out.

Therefore, scaling up, with 10 dishes, it's 10*9=90. So, I think that's solid.

Therefore, the answer is 90. So, I should put that in a box.

**Final Answer**
The number of different combinations of meals Yann and Camille can order is \boxed{90}.

To determine how many different combinations of meals Yann and Camille can order, we need to consider the following:

- There are **10 items** on the menu.
- Each person orders **one dish**.
- They **refuse to order the same dish**.
- The **order matters**, meaning that Yann ordering dish A and Camille ordering dish B is a **different** combination than Yann ordering dish B and Camille ordering dish A.

---

### Step-by-Step Reasoning

1. **Total Possible Choices Without Restriction**:
   - If there were no restrictions, Yann could choose any of the 10 dishes, and Camille could also choose any of the 10 dishes.
   - So, the total number of combinations would be:
     $$
     10 \times 10 = 100
     $$

2. **Subtract the Cases Where They Order the Same Dish**:
   - There are 10 dishes, and for each dish, there is exactly one case where both Yann and Camille order the same dish.
   - So, the number of such cases is:
     $$
     10
     $$
   - Therefore, the number of valid combinations where they **do not** order the same dish is:
     $$
     100 - 10 = 90
     $$

3. **Alternative Approach: Direct Counting**:
   - Yann can choose any of the 10 dishes.
   - Once Yann has chosen a dish, Camille has 9 remaining choices (since she cannot choose the same dish).
   - So, the total number of valid combinations is:
     $$
     10 \times 9 = 90
     $$

4. **Permutations Perspective**:
   - This is equivalent to choosing **2 distinct dishes** from 10 and assigning them to Yann and Camille.
   - The number of **permutations** of 10 items taken 2 at a time is:
     $$
     P(10, 2) = 10 \times 9 = 90
     $$

---

### Final Answer

All approaches confirm that the number of different combinations of meals Yann and Camille can order, under the given conditions, is:

$$
\boxed{90}
$$

[Assistant]
Explanation:
Okay, so Yann and Camille are going to a restaurant, and there are 10 items on the menu. Each of them orders one dish. The question is asking how many different combinations of meals they can order if they refuse to order the same dish. Also, it mentions that it does matter who orders what, so Yann ordering chicken and Camille ordering fish is different from Yann ordering fish and Camille ordering chicken. Alright, let me think through this step by step.

First, let me make sure I understand the problem correctly. There are 10 dishes on the menu. Yann and Camille each choose one dish. They can't choose the same dish. And since it matters who orders what, the order of selection is important. So, for example, if Yann picks dish A and Camille picks dish B, that's a different combination than Yann picking dish B and Camille picking dish A. So, we need to count all such possible ordered pairs where the two dishes are different.

Let me recall some combinatorial principles here. If there were no restrictions, how many combinations would there be? Well, since each person can choose any dish, and their choices are independent, it would be 10 choices for Yann and 10 choices for Camille, so 10*10 = 100 total combinations. But since they can't order the same dish, we need to subtract the cases where they order the same dish.

Alternatively, maybe it's easier to think of it as permutations. Since the order matters, and they can't have the same dish. Wait, but permutations usually involve arranging items where order matters, but here we have two people choosing dishes. Let me think.

Another approach: For Yann, he can choose any of the 10 dishes. Once Yann has chosen a dish, Camille cannot choose that same dish. So, Camille has 9 remaining dishes to choose from. Therefore, the total number of combinations would be 10 * 9 = 90. Is that right?

Wait, let me check. If Yann picks first, there are 10 options. For each of Yann's choices, Camille has 9 options. So, 10*9=90. Alternatively, if we think of it as permutations of 10 dishes taken 2 at a time, which is 10*9=90. Since permutations account for order, which in this case is important because Yann and Camille are distinct individuals. So, that seems to fit.

But let me verify with another approach. Let's think about all possible ordered pairs (Yann's dish, Camille's dish) where the two dishes are different. The total number of ordered pairs is 10*10=100. The number of ordered pairs where they have the same dish is 10 (one for each dish: both choose dish 1, both choose dish 2, etc.). Therefore, subtracting those, we get 100 - 10 = 90. Yep, same answer. So that seems to confirm it.

Alternatively, if I think about combinations versus permutations. If the problem had said that the order doesn't matter, like how many ways can they order two different dishes regardless of who gets which, then it would be combinations. In that case, it would be C(10,2) = 45. But since the problem says that it does matter who orders what, so each of those 45 combinations would correspond to two different ordered pairs. For example, choosing dish A and dish B can be arranged in two ways: Yann has A and Camille has B, or vice versa. Therefore, 45*2=90. Which again gives the same answer. So that's another way to think about it.

So, all three approaches—direct counting, subtracting the same dish cases, and thinking in terms of combinations and then accounting for order—lead me to 90. Therefore, I think the answer is 90.

But let me just make sure there isn't a trick or something I'm missing. The problem says "different combinations of meals". Wait, does "combinations" here mean something different? Wait, in common language, sometimes "combinations" can be ambiguous. But in combinatorics, "combinations" usually refer to unordered selections, while "permutations" are ordered. However, the problem specifically says "It does matter who orders what---Yann ordering chicken and Camille ordering fish is different from Yann ordering fish and Camille ordering chicken." So, they are emphasizing that the order matters. Therefore, the problem is not asking for combinations in the combinatorial sense, but rather permutations. So, the answer is permutations of 10 dishes taken 2 at a time, which is 10*9=90. So, that's consistent.

Alternatively, if they had said "how many different combinations of meals can they order", without specifying that it matters who orders what, then it might be combinations. But since they specifically mention that it does matter, we have to consider the order. Therefore, 90 is correct.

Another way to think about it: For each of the 10 dishes Yann can choose, Camille has 9 options. So, 10*9=90. So, that's straightforward.

Alternatively, if we think of the first person (Yann) has 10 choices, the second person (Camille) has 9 choices, so 10*9=90. So, that's the same as before.

Alternatively, if we think of the problem as assigning dishes to Yann and Camille, with the constraint that they are different. So, the number of injective functions from the set {Yann, Camille} to the set of 10 dishes. The number of injective functions is P(10,2)=10*9=90. So, that's another way to think about it.

Therefore, all these different ways of thinking about it lead me to the same answer. Therefore, I feel confident that the answer is 90.

But just to make sure, let me take a smaller example. Suppose there are 2 dishes on the menu, say A and B. How many combinations? According to the formula, it would be 2*1=2. Let's check. The possibilities are:

Yann orders A, Camille orders B.

Yann orders B, Camille orders A.

That's two combinations, which is correct. If we didn't have the restriction, it would be 2*2=4. Subtract the cases where they order the same dish: 2 cases (both A or both B). So 4-2=2. Correct. So, that works.

Another example: 3 dishes. Then according to formula, 3*2=6. Let's list them:

Yann A, Camille B

Yann A, Camille C

Yann B, Camille A

Yann B, Camille C

Yann C, Camille A

Yann C, Camille B

That's 6, which is correct. If we didn't have the restriction, it would be 9. Subtract 3 cases where they order the same dish, so 6. Correct. So, that seems to check out.

Therefore, scaling up, with 10 dishes, it's 10*9=90. So, I think that's solid.

Therefore, the answer is 90. So, I should put that in a box.

**Final Answer**
The number of different combinations of meals Yann and Camille can order is \boxed{90}.

To determine how many different combinations of meals Yann and Camille can order, we need to consider the following:

- There are **10 items** on the menu.
- Each person orders **one dish**.
- They **refuse to order the same dish**.
- The **order matters**, meaning that Yann ordering dish A and Camille ordering dish B is a **different** combination than Yann ordering dish B and Camille ordering dish A.

---

### Step-by-Step Reasoning

1. **Total Possible Choices Without Restriction**:
   - If there were no restrictions, Yann could choose any of the 10 dishes, and Camille could also choose any of the 10 dishes.
   - So, the total number of combinations would be:
     $$
     10 \times 10 = 100
     $$

2. **Subtract the Cases Where They Order the Same Dish**:
   - There are 10 dishes, and for each dish, there is exactly one case where both Yann and Camille order the same dish.
   - So, the number of such cases is:
     $$
     10
     $$
   - Therefore, the number of valid combinations where they **do not** order the same dish is:
     $$
     100 - 10 = 90
     $$

3. **Alternative Approach: Direct Counting**:
   - Yann can choose any of the 10 dishes.
   - Once Yann has chosen a dish, Camille has 9 remaining choices (since she cannot choose the same dish).
   - So, the total number of valid combinations is:
     $$
     10 \times 9 = 90
     $$

4. **Permutations Perspective**:
   - This is equivalent to choosing **2 distinct dishes** from 10 and assigning them to Yann and Camille.
   - The number of **permutations** of 10 items taken 2 at a time is:
     $$
     P(10, 2) = 10 \times 9 = 90
     $$

---

### Final Answer

All approaches confirm that the number of different combinations of meals Yann and Camille can order, under the given conditions, is:

$$
\boxed{90}
$$

Solution: Yann can order 10 different dishes.  After he has chosen a dish, Camille has 9 choices left for her dish, because she won't order the same dish as Yann did.  Thus there are a total of $10\cdot 9 = \boxed{90}$ different possible combinations of meals.