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 see how I can approach this.

First, I remember that sometimes products of sines can be simplified using trigonometric identities. Maybe I can pair up the terms or use some product-to-sum formulas? Or perhaps there's a symmetry in the angles that can be exploited? Let me list out all the angles:

20°, 40°, 60°, 80°, 100°, 120°, 140°, 160°.

Wait, these angles seem to be spaced by 20°, starting from 20° up to 160°. But 20°, 40°, 60°, 80°, 100°, 120°, 140°, 160°. So that's 8 terms. Let me check: from 20° to 160°, stepping by 20°, yes, that's 8 angles.

But maybe I can use some properties of sine functions. For example, sin(θ) = sin(180° - θ). Let me check which of these angles can be paired with others using this identity.

For example:

sin100° = sin(80°), because 180° - 100° = 80°. Similarly, sin140° = sin(40°), sin160° = sin(20°). Also, sin120° = sin(60°), because 180° - 120° = 60°. Wait, let me verify:

sin(100°) = sin(80°), yes.

sin(140°) = sin(40°), correct.

sin(160°) = sin(20°), yes.

sin(120°) = sin(60°), yes.

So if I rewrite the original product, replacing those terms:

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

Replace sin100° with sin80°, sin140° with sin40°, sin160° with sin20°, sin120° with sin60°. Then the product becomes:

[sin20° sin40° sin60° sin80°] * [sin80° sin60° sin40° sin20°]

So that's [sin20° sin40° sin60° sin80°]^2.

Wait, so the original product is the square of sin20° sin40° sin60° sin80°. That simplifies things a bit. So now I can compute [sin20° sin40° sin60° sin80°]^2.

But maybe even further? Let me check. Let me first compute sin20° sin40° sin60° sin80°.

Alternatively, maybe I can compute the original product by grouping terms differently. Let me check again:

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

Which becomes sin20° sin40° sin60° sin80° sin80° sin60° sin40° sin20° = [sin20° sin40° sin60° sin80°]^2.

Yes, that's correct. So if I can compute sin20° sin40° sin60° sin80°, then square it, that would be the answer.

Alternatively, maybe there is a formula for the product of sines with angles in arithmetic progression? Or maybe use complex numbers? Or perhaps use multiple-angle identities?

Alternatively, since sin60° is known, maybe we can compute that term first.

Let me note that sin60° is √3/2. So if I can compute the product sin20° sin40° sin80°, then multiply by sin60°, then square the whole thing.

But maybe there's a way to compute sin20° sin40° sin80°? Let me think.

Alternatively, I remember that there are formulas for products of sines, for example:

Product of sines: For example, sinθ sin(60° - θ) sin(60° + θ) = (sin3θ)/4.

Wait, let me check that identity.

Yes, there is an identity that says:

sinθ sin(60° - θ) sin(60° + θ) = (sin3θ)/4.

So if I can apply that here. Let me see.

But in our case, we have sin20° sin40° sin80°. Let me see if these angles can be represented in that form.

Let me take θ = 20°, then 60° - θ = 40°, 60° + θ = 80°. Exactly! So θ = 20°, so sin20° sin40° sin80° = sin3θ /4 = sin60° / 4.

Since 3θ = 60°, so sin60° is √3/2. Therefore, sin20° sin40° sin80° = (√3/2)/4 = √3 / 8.

Wait, so that product is √3 / 8? Let me confirm.

Let me compute sin20° sin40° sin80°.

Alternatively, let me use another method to verify.

Let me compute sin20° sin40° sin80°.

Alternatively, using the identity I mentioned:

sinθ sin(60° - θ) sin(60° + θ) = (sin3θ)/4.

Yes, so substituting θ=20°, we get sin20° sin40° sin80° = sin60° / 4 = (√3/2)/4 = √3/8.

So that's correct.

Therefore, returning to our original product.

We have [sin20° sin40° sin60° sin80°]^2.

Which is [sin20° sin40° sin80° * sin60°]^2 = [(√3/8) * sin60°]^2.

Wait, sin60° is √3/2. Therefore:

[(√3/8) * (√3/2)]^2.

Let me compute that:

First, multiply √3/8 and √3/2.

√3 * √3 = 3, denominators 8*2 = 16.

So that's 3/16.

Then square that: (3/16)^2 = 9/256.

Wait, so the entire original product would be 9/256?

But let me double-check.

Original product is [sin20° sin40° sin60° sin80°]^2.

But sin20° sin40° sin80° is √3/8, and sin60° is √3/2. Therefore, multiplying those together: (√3/8) * (√3/2) = (3)/16, then squaring that gives 9/256.

But let me check whether I have considered all the terms correctly. Wait, the original product had [sin20° sin40° sin60° sin80°]^2. Wait, but sin60° was part of the original product? Let me check again.

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

After using sin100° = sin80°, sin140°=sin40°, sin160°=sin20°, sin120°=sin60°, the product becomes:

sin20° sin40° sin60° sin80° sin80° sin60° sin40° sin20° = (sin20° sin40° sin60° sin80°)^2.

Yes. Therefore, sin20° sin40° sin60° sin80° is equal to [sin20° sin40° sin80°] * sin60°, which we found as (√3/8) * (√3/2) = 3/16. Then squaring that gives 9/256.

But wait, let me check with another approach. Maybe there's an error here? Let me check step by step again.

First, the product sin20° sin40° sin60° sin80° sin100° sin120° sin140° sin160°.

As we have sin100°=sin80°, sin140°=sin40°, sin160°=sin20°, sin120°=sin60°, so substituting:

sin20° sin40° sin60° sin80° sin80° sin60° sin40° sin20°.

Which is [sin20° sin40° sin60° sin80°]^2.

Yes. Therefore, if we can compute sin20° sin40° sin60° sin80°, then square it.

But earlier, we computed sin20° sin40° sin80° = √3 / 8. Then multiplying by sin60° = √3 / 2 gives √3 / 8 * √3 / 2 = 3 / 16. Then squaring that gives 9 / 256.

But maybe there is another way to compute sin20° sin40° sin60° sin80°? Let me try to compute it step by step.

Alternatively, use product formulas.

Alternatively, note that sinθ sin(60° - θ) sin(60° + θ) = sin3θ / 4.

But we already used that for sin20° sin40° sin80° = sin60° / 4 = √3 / 8.

But then multiplied by sin60°, so √3 / 8 * √3 / 2 = 3 / 16.

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

Hmm. But let me check if that answer is correct. Maybe I can compute the product numerically.

Let me calculate the original product numerically, and check if it's equal to 9/256. Let me compute each sine term:

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

Convert degrees to decimal for calculation:

sin20° ≈ 0.3420

sin40° ≈ 0.6428

sin60° ≈ 0.8660

sin80° ≈ 0.9848

sin100° = sin(80°) ≈ 0.9848

sin120° = sin(60°) ≈ 0.8660

sin140° = sin(40°) ≈ 0.6428

sin160° = sin(20°) ≈ 0.3420

So multiplying all together:

0.3420 * 0.6428 * 0.8660 * 0.9848 * 0.9848 * 0.8660 * 0.6428 * 0.3420

Let me compute step by step:

First pair them as [0.3420 * 0.6428] * [0.8660 * 0.9848] * [0.9848 * 0.8660] * [0.6428 * 0.3420]

Wait, perhaps grouping them as [sin20° sin160°] * [sin40° sin140°] * [sin60° sin120°] * [sin80° sin100°]

Which would be [sin20° sin20°] * [sin40° sin40°] * [sin60° sin60°] * [sin80° sin80°] because sin160°=sin20°, etc.

So that would be [sin^2 20° * sin^2 40° * sin^2 60° * sin^2 80°]. Therefore, the original product is [sin20° sin40° sin60° sin80°]^2, which matches our earlier conclusion.

Calculating each squared term:

sin^2 20° ≈ (0.3420)^2 ≈ 0.116964

sin^2 40° ≈ (0.6428)^2 ≈ 0.4132

sin^2 60° ≈ (0.8660)^2 ≈ 0.75

sin^2 80° ≈ (0.9848)^2 ≈ 0.9698

Multiply all these together:

0.116964 * 0.4132 * 0.75 * 0.9698

First compute 0.116964 * 0.4132 ≈ let's see:

Approximately 0.116964 * 0.4 = 0.0467856, plus 0.116964 * 0.0132 ≈ 0.001544, so total ≈ 0.0467856 + 0.001544 ≈ 0.0483296

Then multiply by 0.75: 0.0483296 * 0.75 ≈ 0.0362472

Then multiply by 0.9698 ≈ 0.0362472 * 0.9698 ≈

Let me compute 0.0362472 * 0.9698 ≈

First, 0.0362472 * 1 = 0.0362472

Minus 0.0362472 * (1 - 0.9698) = 0.0362472 * 0.0302 ≈

Calculate 0.0362472 * 0.03 = 0.0010874 and 0.0362472 * 0.0002 ≈ 0.0000072. Total ≈ 0.0010946.

Therefore, approximate value is 0.0362472 - 0.0010946 ≈ 0.0351526.

So the total product is approximately 0.03515.

Now let's compute 9/256.

Calculate 9 divided by 256.

Well, 256 / 9 ≈ 28.444... so 1/256 ≈ 0.00390625, so 9 * 0.00390625 = 0.03515625.

Wow, that's very close to the approximate value we calculated (≈0.03515). Therefore, that gives me confidence that the exact value is 9/256.

Therefore, the answer is 9/256.

Wait, but let me check once more. Because when I calculated [sin20° sin40° sin60° sin80°]^2, I first found that sin20° sin40° sin80° = √3 / 8, then multiplied by sin60° = √3 / 2 to get 3 / 16. Then squared it to get 9 / 256.

But let me check the step where I used the identity.

The identity says sinθ sin(60° - θ) sin(60° + θ) = sin3θ / 4.

So if θ = 20°, then 3θ = 60°, so sin60° / 4 = (√3/2)/4 = √3 / 8.

Therefore, sin20° sin40° sin80° = √3 / 8.

Then multiplying by sin60° gives √3 / 8 * √3 / 2 = (3)/16.

Then squaring that gives 9 / 256.

Yes, that's correct. Therefore, the answer is 9/256.

Alternatively, maybe there is another approach, such as using complex numbers or multiple-angle formulas?

Alternatively, using the identity for product of sines.

But given that the numerical calculation matches 9/256 ≈ 0.03515625, which is very close to the approximate product I computed earlier (~0.03515), I think that this is correct.

But just to make sure, perhaps I can check another way. Let me try to compute the original product step by step using another identity.

Alternatively, note that the product sinθ sin(60° - θ) sin(60° + θ) = sin3θ / 4.

So if I have multiple terms, maybe I can apply this multiple times?

Alternatively, think about the entire product.

Wait, the original product after pairing is [sin20° sin40° sin60° sin80°]^2. Let me think if there's another way to compute this.

Alternatively, use the identity for product of sines:

There is a formula for the product of sines of angles in arithmetic progression?

Alternatively, perhaps use complex exponentials?

But that might be more complicated. However, let me try.

But since the earlier method gave me 9/256, which is confirmed numerically, I think that's solid.

Alternatively, let me check if there's an error in pairing.

Wait, the original product is 8 terms. When I substituted, I got [sin20° sin40° sin60° sin80°]^2. Which is 4 terms squared.

But let me count again:

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

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

Therefore, the product becomes:

sin20° * sin40° * sin60° * sin80° * sin80° * sin60° * sin40° * sin20°.

Which is indeed [sin20° sin40° sin60° sin80°]^2.

Yes. So that's correct.

Therefore, the answer is 9/256.

Alternatively, let me check with another identity. Let me recall that the product of sines from k=1 to n of sin(kπ/(2n+1)) has a known formula.

But in this case, our angles are 20°, 40°, ..., 160°, which are multiples of 20°, which is π/9 radians.

Wait, 20° is π/9, since 180° is π, so 20° is π/9. Similarly, 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 the product from k=1 to 8 of sin(kπ/9). Wait, but let me check:

Wait, kπ/9 for k=1 to 8 would be π/9, 2π/9, ..., 8π/9. Which is exactly the angles: 20°, 40°, ..., 160°, since π radians is 180°, so π/9 is 20°, etc.

Therefore, the original product is the product_{k=1}^8 sin(kπ/9).

Now, there is a known formula for the product of sines of the form product_{k=1}^{n-1} sin(kπ/n) = n / 2^{n-1}.

Wait, but here n would be 9? Wait, if n is 9, then product_{k=1}^{8} sin(kπ/9) = 9 / 2^8 = 9 / 256.

Wait, exactly!

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

But wait, let me check.

Actually, the formula is that product_{k=1}^{n-1} sin(kπ/n) = n / 2^{n-1}.

But wait, let me check for n=3: product_{k=1}^2 sin(kπ/3) = sin(π/3) sin(2π/3) = (√3/2)(√3/2) = 3/4. And n / 2^{n-1} = 3 / 2^{2} = 3/4. Correct.

For n=4: product_{k=1}^3 sin(kπ/4) = sin(π/4) sin(π/2) sin(3π/4) = (√2/2)(1)(√2/2) = (2/4)(1) = 1/2. And n / 2^{n-1} = 4 / 8 = 1/2. Correct.

Therefore, for n=9, the product_{k=1}^8 sin(kπ/9) = 9 / 2^8 = 9 / 256.

Which is exactly our product! Therefore, the answer is 9/256.

Therefore, this confirms that the product is indeed 9/256, and that our previous calculation was correct.

Therefore, the answer is 9/256.

So, to recap:

Original product is product_{k=1}^8 sin(kπ/9) = 9 / 2^8 = 9 / 256.

Therefore, the value is 9/256.

But just to make sure, let me check the formula again.

Yes, according to the formula, product_{k=1}^{n-1} sin(kπ/n) = n / 2^{n-1}.

So for n=9, it's 9 / 2^8 = 9 / 256.

Therefore, this is a direct application of the formula.

Therefore, this gives me even more confidence that the answer is indeed 9/256.

So, the answer is \boxed{\dfrac{9}{256}}.

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

To evaluate the expression:

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

we begin by observing the symmetry and identities in trigonometry that allow for simplification.

---

### Step 1: Use the identity $\sin(180^\circ - \theta) = \sin\theta$

Apply this to each of the angles beyond 90°:

- $\sin 100^\circ = \sin(180^\circ - 80^\circ) = \sin 80^\circ$
- $\sin 120^\circ = \sin(180^\circ - 60^\circ) = \sin 60^\circ$
- $\sin 140^\circ = \sin(180^\circ - 40^\circ) = \sin 40^\circ$
- $\sin 160^\circ = \sin(180^\circ - 20^\circ) = \sin 20^\circ$

Substituting these into the product:

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

This simplifies to:

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

Now we need to compute:

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

---

### Step 2: Use the identity for product of sines

There is a known identity:

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

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 with the remaining factor, $\sin 60^\circ = \frac{\sqrt{3}}{2}$:

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

Then square this result to get the original product:

$$
\left(\frac{3}{16}\right)^2 = \frac{9}{256}
$$

---

### Step 3: Alternative confirmation using known formula

Another powerful confirmation comes from the identity for the product of sines of angles in arithmetic progression:

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

In this case, the angles are:

$$
20^\circ, 40^\circ, \dots, 160^\circ
\Rightarrow \frac{\pi}{9}, \frac{2\pi}{9}, \dots, \frac{8\pi}{9}
$$

So, $n = 9$, and the formula gives:

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

This matches our result from the earlier approach.

---

### 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 see how I can approach this.

First, I remember that sometimes products of sines can be simplified using trigonometric identities. Maybe I can pair up the terms or use some product-to-sum formulas? Or perhaps there's a symmetry in the angles that can be exploited? Let me list out all the angles:

20°, 40°, 60°, 80°, 100°, 120°, 140°, 160°.

Wait, these angles seem to be spaced by 20°, starting from 20° up to 160°. But 20°, 40°, 60°, 80°, 100°, 120°, 140°, 160°. So that's 8 terms. Let me check: from 20° to 160°, stepping by 20°, yes, that's 8 angles.

But maybe I can use some properties of sine functions. For example, sin(θ) = sin(180° - θ). Let me check which of these angles can be paired with others using this identity.

For example:

sin100° = sin(80°), because 180° - 100° = 80°. Similarly, sin140° = sin(40°), sin160° = sin(20°). Also, sin120° = sin(60°), because 180° - 120° = 60°. Wait, let me verify:

sin(100°) = sin(80°), yes.

sin(140°) = sin(40°), correct.

sin(160°) = sin(20°), yes.

sin(120°) = sin(60°), yes.

So if I rewrite the original product, replacing those terms:

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

Replace sin100° with sin80°, sin140° with sin40°, sin160° with sin20°, sin120° with sin60°. Then the product becomes:

[sin20° sin40° sin60° sin80°] * [sin80° sin60° sin40° sin20°]

So that's [sin20° sin40° sin60° sin80°]^2.

Wait, so the original product is the square of sin20° sin40° sin60° sin80°. That simplifies things a bit. So now I can compute [sin20° sin40° sin60° sin80°]^2.

But maybe even further? Let me check. Let me first compute sin20° sin40° sin60° sin80°.

Alternatively, maybe I can compute the original product by grouping terms differently. Let me check again:

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

Which becomes sin20° sin40° sin60° sin80° sin80° sin60° sin40° sin20° = [sin20° sin40° sin60° sin80°]^2.

Yes, that's correct. So if I can compute sin20° sin40° sin60° sin80°, then square it, that would be the answer.

Alternatively, maybe there is a formula for the product of sines with angles in arithmetic progression? Or maybe use complex numbers? Or perhaps use multiple-angle identities?

Alternatively, since sin60° is known, maybe we can compute that term first.

Let me note that sin60° is √3/2. So if I can compute the product sin20° sin40° sin80°, then multiply by sin60°, then square the whole thing.

But maybe there's a way to compute sin20° sin40° sin80°? Let me think.

Alternatively, I remember that there are formulas for products of sines, for example:

Product of sines: For example, sinθ sin(60° - θ) sin(60° + θ) = (sin3θ)/4.

Wait, let me check that identity.

Yes, there is an identity that says:

sinθ sin(60° - θ) sin(60° + θ) = (sin3θ)/4.

So if I can apply that here. Let me see.

But in our case, we have sin20° sin40° sin80°. Let me see if these angles can be represented in that form.

Let me take θ = 20°, then 60° - θ = 40°, 60° + θ = 80°. Exactly! So θ = 20°, so sin20° sin40° sin80° = sin3θ /4 = sin60° / 4.

Since 3θ = 60°, so sin60° is √3/2. Therefore, sin20° sin40° sin80° = (√3/2)/4 = √3 / 8.

Wait, so that product is √3 / 8? Let me confirm.

Let me compute sin20° sin40° sin80°.

Alternatively, let me use another method to verify.

Let me compute sin20° sin40° sin80°.

Alternatively, using the identity I mentioned:

sinθ sin(60° - θ) sin(60° + θ) = (sin3θ)/4.

Yes, so substituting θ=20°, we get sin20° sin40° sin80° = sin60° / 4 = (√3/2)/4 = √3/8.

So that's correct.

Therefore, returning to our original product.

We have [sin20° sin40° sin60° sin80°]^2.

Which is [sin20° sin40° sin80° * sin60°]^2 = [(√3/8) * sin60°]^2.

Wait, sin60° is √3/2. Therefore:

[(√3/8) * (√3/2)]^2.

Let me compute that:

First, multiply √3/8 and √3/2.

√3 * √3 = 3, denominators 8*2 = 16.

So that's 3/16.

Then square that: (3/16)^2 = 9/256.

Wait, so the entire original product would be 9/256?

But let me double-check.

Original product is [sin20° sin40° sin60° sin80°]^2.

But sin20° sin40° sin80° is √3/8, and sin60° is √3/2. Therefore, multiplying those together: (√3/8) * (√3/2) = (3)/16, then squaring that gives 9/256.

But let me check whether I have considered all the terms correctly. Wait, the original product had [sin20° sin40° sin60° sin80°]^2. Wait, but sin60° was part of the original product? Let me check again.

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

After using sin100° = sin80°, sin140°=sin40°, sin160°=sin20°, sin120°=sin60°, the product becomes:

sin20° sin40° sin60° sin80° sin80° sin60° sin40° sin20° = (sin20° sin40° sin60° sin80°)^2.

Yes. Therefore, sin20° sin40° sin60° sin80° is equal to [sin20° sin40° sin80°] * sin60°, which we found as (√3/8) * (√3/2) = 3/16. Then squaring that gives 9/256.

But wait, let me check with another approach. Maybe there's an error here? Let me check step by step again.

First, the product sin20° sin40° sin60° sin80° sin100° sin120° sin140° sin160°.

As we have sin100°=sin80°, sin140°=sin40°, sin160°=sin20°, sin120°=sin60°, so substituting:

sin20° sin40° sin60° sin80° sin80° sin60° sin40° sin20°.

Which is [sin20° sin40° sin60° sin80°]^2.

Yes. Therefore, if we can compute sin20° sin40° sin60° sin80°, then square it.

But earlier, we computed sin20° sin40° sin80° = √3 / 8. Then multiplying by sin60° = √3 / 2 gives √3 / 8 * √3 / 2 = 3 / 16. Then squaring that gives 9 / 256.

But maybe there is another way to compute sin20° sin40° sin60° sin80°? Let me try to compute it step by step.

Alternatively, use product formulas.

Alternatively, note that sinθ sin(60° - θ) sin(60° + θ) = sin3θ / 4.

But we already used that for sin20° sin40° sin80° = sin60° / 4 = √3 / 8.

But then multiplied by sin60°, so √3 / 8 * √3 / 2 = 3 / 16.

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

Hmm. But let me check if that answer is correct. Maybe I can compute the product numerically.

Let me calculate the original product numerically, and check if it's equal to 9/256. Let me compute each sine term:

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

Convert degrees to decimal for calculation:

sin20° ≈ 0.3420

sin40° ≈ 0.6428

sin60° ≈ 0.8660

sin80° ≈ 0.9848

sin100° = sin(80°) ≈ 0.9848

sin120° = sin(60°) ≈ 0.8660

sin140° = sin(40°) ≈ 0.6428

sin160° = sin(20°) ≈ 0.3420

So multiplying all together:

0.3420 * 0.6428 * 0.8660 * 0.9848 * 0.9848 * 0.8660 * 0.6428 * 0.3420

Let me compute step by step:

First pair them as [0.3420 * 0.6428] * [0.8660 * 0.9848] * [0.9848 * 0.8660] * [0.6428 * 0.3420]

Wait, perhaps grouping them as [sin20° sin160°] * [sin40° sin140°] * [sin60° sin120°] * [sin80° sin100°]

Which would be [sin20° sin20°] * [sin40° sin40°] * [sin60° sin60°] * [sin80° sin80°] because sin160°=sin20°, etc.

So that would be [sin^2 20° * sin^2 40° * sin^2 60° * sin^2 80°]. Therefore, the original product is [sin20° sin40° sin60° sin80°]^2, which matches our earlier conclusion.

Calculating each squared term:

sin^2 20° ≈ (0.3420)^2 ≈ 0.116964

sin^2 40° ≈ (0.6428)^2 ≈ 0.4132

sin^2 60° ≈ (0.8660)^2 ≈ 0.75

sin^2 80° ≈ (0.9848)^2 ≈ 0.9698

Multiply all these together:

0.116964 * 0.4132 * 0.75 * 0.9698

First compute 0.116964 * 0.4132 ≈ let's see:

Approximately 0.116964 * 0.4 = 0.0467856, plus 0.116964 * 0.0132 ≈ 0.001544, so total ≈ 0.0467856 + 0.001544 ≈ 0.0483296

Then multiply by 0.75: 0.0483296 * 0.75 ≈ 0.0362472

Then multiply by 0.9698 ≈ 0.0362472 * 0.9698 ≈

Let me compute 0.0362472 * 0.9698 ≈

First, 0.0362472 * 1 = 0.0362472

Minus 0.0362472 * (1 - 0.9698) = 0.0362472 * 0.0302 ≈

Calculate 0.0362472 * 0.03 = 0.0010874 and 0.0362472 * 0.0002 ≈ 0.0000072. Total ≈ 0.0010946.

Therefore, approximate value is 0.0362472 - 0.0010946 ≈ 0.0351526.

So the total product is approximately 0.03515.

Now let's compute 9/256.

Calculate 9 divided by 256.

Well, 256 / 9 ≈ 28.444... so 1/256 ≈ 0.00390625, so 9 * 0.00390625 = 0.03515625.

Wow, that's very close to the approximate value we calculated (≈0.03515). Therefore, that gives me confidence that the exact value is 9/256.

Therefore, the answer is 9/256.

Wait, but let me check once more. Because when I calculated [sin20° sin40° sin60° sin80°]^2, I first found that sin20° sin40° sin80° = √3 / 8, then multiplied by sin60° = √3 / 2 to get 3 / 16. Then squared it to get 9 / 256.

But let me check the step where I used the identity.

The identity says sinθ sin(60° - θ) sin(60° + θ) = sin3θ / 4.

So if θ = 20°, then 3θ = 60°, so sin60° / 4 = (√3/2)/4 = √3 / 8.

Therefore, sin20° sin40° sin80° = √3 / 8.

Then multiplying by sin60° gives √3 / 8 * √3 / 2 = (3)/16.

Then squaring that gives 9 / 256.

Yes, that's correct. Therefore, the answer is 9/256.

Alternatively, maybe there is another approach, such as using complex numbers or multiple-angle formulas?

Alternatively, using the identity for product of sines.

But given that the numerical calculation matches 9/256 ≈ 0.03515625, which is very close to the approximate product I computed earlier (~0.03515), I think that this is correct.

But just to make sure, perhaps I can check another way. Let me try to compute the original product step by step using another identity.

Alternatively, note that the product sinθ sin(60° - θ) sin(60° + θ) = sin3θ / 4.

So if I have multiple terms, maybe I can apply this multiple times?

Alternatively, think about the entire product.

Wait, the original product after pairing is [sin20° sin40° sin60° sin80°]^2. Let me think if there's another way to compute this.

Alternatively, use the identity for product of sines:

There is a formula for the product of sines of angles in arithmetic progression?

Alternatively, perhaps use complex exponentials?

But that might be more complicated. However, let me try.

But since the earlier method gave me 9/256, which is confirmed numerically, I think that's solid.

Alternatively, let me check if there's an error in pairing.

Wait, the original product is 8 terms. When I substituted, I got [sin20° sin40° sin60° sin80°]^2. Which is 4 terms squared.

But let me count again:

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

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

Therefore, the product becomes:

sin20° * sin40° * sin60° * sin80° * sin80° * sin60° * sin40° * sin20°.

Which is indeed [sin20° sin40° sin60° sin80°]^2.

Yes. So that's correct.

Therefore, the answer is 9/256.

Alternatively, let me check with another identity. Let me recall that the product of sines from k=1 to n of sin(kπ/(2n+1)) has a known formula.

But in this case, our angles are 20°, 40°, ..., 160°, which are multiples of 20°, which is π/9 radians.

Wait, 20° is π/9, since 180° is π, so 20° is π/9. Similarly, 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 the product from k=1 to 8 of sin(kπ/9). Wait, but let me check:

Wait, kπ/9 for k=1 to 8 would be π/9, 2π/9, ..., 8π/9. Which is exactly the angles: 20°, 40°, ..., 160°, since π radians is 180°, so π/9 is 20°, etc.

Therefore, the original product is the product_{k=1}^8 sin(kπ/9).

Now, there is a known formula for the product of sines of the form product_{k=1}^{n-1} sin(kπ/n) = n / 2^{n-1}.

Wait, but here n would be 9? Wait, if n is 9, then product_{k=1}^{8} sin(kπ/9) = 9 / 2^8 = 9 / 256.

Wait, exactly!

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

But wait, let me check.

Actually, the formula is that product_{k=1}^{n-1} sin(kπ/n) = n / 2^{n-1}.

But wait, let me check for n=3: product_{k=1}^2 sin(kπ/3) = sin(π/3) sin(2π/3) = (√3/2)(√3/2) = 3/4. And n / 2^{n-1} = 3 / 2^{2} = 3/4. Correct.

For n=4: product_{k=1}^3 sin(kπ/4) = sin(π/4) sin(π/2) sin(3π/4) = (√2/2)(1)(√2/2) = (2/4)(1) = 1/2. And n / 2^{n-1} = 4 / 8 = 1/2. Correct.

Therefore, for n=9, the product_{k=1}^8 sin(kπ/9) = 9 / 2^8 = 9 / 256.

Which is exactly our product! Therefore, the answer is 9/256.

Therefore, this confirms that the product is indeed 9/256, and that our previous calculation was correct.

Therefore, the answer is 9/256.

So, to recap:

Original product is product_{k=1}^8 sin(kπ/9) = 9 / 2^8 = 9 / 256.

Therefore, the value is 9/256.

But just to make sure, let me check the formula again.

Yes, according to the formula, product_{k=1}^{n-1} sin(kπ/n) = n / 2^{n-1}.

So for n=9, it's 9 / 2^8 = 9 / 256.

Therefore, this is a direct application of the formula.

Therefore, this gives me even more confidence that the answer is indeed 9/256.

So, the answer is \boxed{\dfrac{9}{256}}.

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

To evaluate the expression:

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

we begin by observing the symmetry and identities in trigonometry that allow for simplification.

---

### Step 1: Use the identity $\sin(180^\circ - \theta) = \sin\theta$

Apply this to each of the angles beyond 90°:

- $\sin 100^\circ = \sin(180^\circ - 80^\circ) = \sin 80^\circ$
- $\sin 120^\circ = \sin(180^\circ - 60^\circ) = \sin 60^\circ$
- $\sin 140^\circ = \sin(180^\circ - 40^\circ) = \sin 40^\circ$
- $\sin 160^\circ = \sin(180^\circ - 20^\circ) = \sin 20^\circ$

Substituting these into the product:

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

This simplifies to:

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

Now we need to compute:

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

---

### Step 2: Use the identity for product of sines

There is a known identity:

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

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 with the remaining factor, $\sin 60^\circ = \frac{\sqrt{3}}{2}$:

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

Then square this result to get the original product:

$$
\left(\frac{3}{16}\right)^2 = \frac{9}{256}
$$

---

### Step 3: Alternative confirmation using known formula

Another powerful confirmation comes from the identity for the product of sines of angles in arithmetic progression:

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

In this case, the angles are:

$$
20^\circ, 40^\circ, \dots, 160^\circ
\Rightarrow \frac{\pi}{9}, \frac{2\pi}{9}, \dots, \frac{8\pi}{9}
$$

So, $n = 9$, and the formula gives:

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

This matches our result from the earlier approach.

---

### 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 equals 36. The planes are x - z = 0, x - 2y + z = 0, and x + y + z = 0.

First, let me 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 wait, the given planes are already in the form ax + by + cz = 0. So for each plane, the distance formula simplifies. Let me write down each distance squared.

Let me denote the three planes as follows:

Plane 1: x - z = 0. So, coefficients are a1 = 1, b1 = 0, c1 = -1, d1 = 0. Wait, but the general formula is ax + by + cz + d = 0, so for plane 1, it's x - z = 0, so d is 0. Therefore, the distance from P = (x, y, z) to plane 1 is |x - z| / sqrt(1² + 0² + (-1)²) = |x - z| / sqrt(2). Therefore, d1² = (x - z)² / 2.

Similarly for the other planes. Let me do the same for plane 2: x - 2y + z = 0. So coefficients a2 = 1, b2 = -2, c2 = 1, d2 = 0. Therefore, distance d2 = |x - 2y + z| / sqrt(1 + 4 + 1) = |x - 2y + z| / sqrt(6). Thus, d2² = (x - 2y + z)² / 6.

Plane 3: x + y + z = 0. Coefficients a3 = 1, b3 = 1, c3 = 1, d3 = 0. Therefore, distance d3 = |x + y + z| / sqrt(1 + 1 + 1) = |x + y + z| / sqrt(3). Hence, d3² = (x + y + z)² / 3.

So the equation given is d1² + d2² + d3² = 36. Therefore, substituting the expressions above:

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

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

Let me note that denominators are 2, 6, 3. To combine them, it might be easier to find a common denominator. Let me multiply each term by 6 to eliminate denominators:

Multiply entire equation by 6:

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

So now the equation becomes:

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 take them one by one.

First term: 3(x - z)². Let's expand (x - z)²: x² - 2xz + z². Multiply by 3: 3x² - 6xz + 3z².

Second term: (x - 2y + z)². Let me expand that. Let me denote it as [(x + z) - 2y]^2. Which is (x + z)² - 4y(x + z) + 4y². Let me compute that:

(x + z)^2 = x² + 2xz + z². Then -4y(x + z) = -4xy -4yz. Then +4y². So altogether:

x² + 2xz + z² - 4xy - 4yz + 4y².

Third term: 2(x + y + z)^2. First expand (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 + 2xz + z² - 4xy - 4yz + 4y²

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

Adding all together:

Let me compute term by term.

For x²: 3x² (from first term) + x² (second term) + 2x² (third term) = 3 + 1 + 2 = 6x².

For y²: 0 (from first term) + 4y² (second term) + 2y² (third term) = 6y².

For z²: 3z² (from first term) + z² (second term) + 2z² (third term) = 6z².

Now cross terms:

For xy: -4xy (from second term) + 4xy (from third term) = 0.

For xz: -6xz (from first term) + 2xz (from second term) + 4xz (from third term) = (-6 + 2 + 4)xz = 0xz.

For yz: -4yz (from second term) + 4yz (from third term) = 0.

Wait, so all cross terms cancel out? That leaves only the squared terms:

6x² + 6y² + 6z² = 216.

So that simplifies to 6(x² + y² + z²) = 216, divide both sides by 6:

x² + y² + z² = 36.

Wait, so the equation reduces to x² + y² + z² = 36. That is, a sphere with radius 6 centered at the origin?

But let me check my steps again because that seems surprisingly simple. Let me verify the expansion again step by step.

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

Second term: (x - 2y + z)^2. Let me expand it as (x + z - 2y)^2. Let me compute (a - b)^2 where a = x + z and b = 2y. Then it's a² - 2ab + b². So (x + z)^2 - 4y(x + z) + 4y². Which is x² + 2xz + z² - 4xy - 4yz + 4y². 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 together:

First term: 3x² -6xz +3z²

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

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

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

y² terms: 4y² (from second term) + 2y² (from third term) = 6y². Correct.

z² terms: 3z² (from first term) + z² (from second term) + 2z² (from third term) = 6z². Correct.

xy terms: -4xy (from second term) +4xy (from third term) = 0. Correct.

xz terms: -6xz (from first term) + 2xz (from second term) +4xz (from third term) = (-6 + 2 + 4)xz = 0. Correct.

yz terms: -4yz (from second term) +4yz (from third term) = 0. Correct.

So indeed, all cross terms cancel out, leaving 6x² + 6y² + 6z² = 216 => x² + y² + z² = 36.

Therefore, the equation d1² + d2² + d3² = 36 simplifies to the sphere of radius 6 centered at the origin. Therefore, the set S is a sphere with radius 6, so the volume enclosed by S is the volume of this sphere.

Wait, but the problem says "Find the region of the volume enclosed by S." So since S is a sphere, the volume enclosed is just the volume of the sphere with radius 6.

But let me think again. Wait, is there something wrong here? Because the original question mentions distances to three planes, and then through simplification, we get a sphere. However, maybe there is a mistake here? Let me check with a specific point.

For example, take the origin (0, 0, 0). Let me compute d1² + d2² + d3².

For the origin:

d1 is distance to x - z = 0: |0 - 0| / sqrt(2) = 0. So d1² = 0.

Similarly, d2 is |0 - 0 + 0| / sqrt(6) = 0, d2² = 0.

d3 is |0 + 0 + 0| / sqrt(3) = 0. So sum is 0, which is less than 36. So origin is inside S.

Now take a point on the sphere x² + y² + z² = 36. Let me take (6, 0, 0). Compute d1² + d2² + d3².

For point (6, 0, 0):

d1² = (6 - 0)^2 / 2 = 36 / 2 = 18.

d2² = (6 - 0 + 0)^2 / 6 = 36 / 6 = 6.

d3² = (6 + 0 + 0)^2 / 3 = 36 / 3 = 12.

Sum: 18 + 6 + 12 = 36. Correct. So the point (6, 0, 0) is on S. Similarly, any point on x² + y² + z² = 36 will satisfy the equation. Therefore, our derivation seems correct.

Hence, the set S is indeed the sphere of radius 6 centered at the origin, so the volume enclosed by S is (4/3)πr³ = (4/3)π*(6)^3 = (4/3)π*216 = 288π.

But the problem says "Find the region of the volume enclosed by S." So the answer is the volume of the sphere, which is 288π. However, let me check if there's some detail I missed. For example, maybe the distances are signed? But no, the distance is always non-negative, so squared distances are positive. So the equation is just the sphere.

Alternatively, maybe there's some constraint? But no, the algebra shows that the equation reduces to the sphere. Hence, the region is a sphere with radius 6, and its volume is 288π.

Wait, but let me check with another point. Let me take (0, 0, 6). Then:

d1² = (0 - 6)^2 / 2 = 36/2 = 18.

d2² = (0 - 0 + 6)^2 /6 = 36/6 = 6.

d3² = (0 + 0 + 6)^2 / 3 = 36/3 = 12.

Sum is 18 + 6 + 12 = 36. Correct. Similarly for (0, 6, 0):

d1² = (0 - 0)^2 /2 = 0.

d2² = (0 - 2*6 + 0)^2 /6 = (-12)^2 /6 = 144 /6 = 24.

d3² = (0 + 6 + 0)^2 /3 = 36 /3 = 12. Sum is 0 + 24 + 12 = 36. Correct. So yes, it works.

Therefore, the conclusion is that the set S is a sphere of radius 6, so the volume enclosed is 4/3 π * 6³ = 288 π. Therefore, the answer should be \boxed{288\pi}.

Wait, but let me check once again. The problem says "the region of the volume enclosed by S." So S is the set of points P such that d1² + d2² + d3² = 36. Which we found is the sphere x² + y² + z² = 36. Therefore, the volume inside S would be the volume of the sphere. However, is there a possibility that S is not the entire sphere? Because sometimes equations can have constraints.

But in our case, we started with d1² + d2² + d3² = 36, which translated to x² + y² + z² = 36. So S is the sphere itself, so the region enclosed by S is the interior of the sphere. But wait, the problem says "the set of points P such that d1² + d2² + d3² = 36", which is the sphere itself. However, the volume enclosed by S would be the set of points inside the sphere. Wait, but actually, the wording is a bit ambiguous. Sometimes, when they say "the region of the volume enclosed by S", they might mean the set S itself. But since S is a sphere, which is a surface, not a volume. However, the problem says "the set of points P such that d1² + d2² + d3² = 36", which is the sphere, but the question says "the region of the volume enclosed by S." So that suggests that S is the boundary of the region, which is a sphere, so the volume enclosed is the interior of the sphere.

But then, in that case, the answer is the volume of the sphere with radius 6, which is 288π. However, let me check again.

Alternatively, maybe there is some mistake here. Let me think again. Because the equation we derived is d1² + d2² + d3² = 36, which is equivalent to x² + y² + z² = 36. Therefore, the set S is the sphere of radius 6. Therefore, the volume enclosed by S would be the set of points inside or on the sphere. However, the problem says "the region of the volume enclosed by S", so if S is the sphere, then the volume enclosed is the ball of radius 6. Therefore, the answer is 4/3 π (6)^3 = 288 π.

Alternatively, maybe there's a misunderstanding here. For example, maybe the problem is not in 3D? But no, it's talking about points P in 3D space, with three planes. So it's definitely 3D.

Alternatively, perhaps I made an error in translating the distances. Let me recheck the distance formulas.

For plane x - z = 0, the distance from (x,y,z) is |x - z| / sqrt(1^2 + 0^2 + (-1)^2) = |x - z| / sqrt(2). Squared is (x - z)^2 / 2. Correct. Then same for other planes.

Then when we expanded the equation, all cross terms canceled out, leading to x² + y² + z² = 36. Therefore, the conclusion is correct.

Alternatively, maybe the problem is in another coordinate system or something? But no.

Therefore, I think the answer is 288π. Therefore, the volume enclosed by S is 288π. Therefore, the answer is \boxed{288\pi}.

**Final Answer**
\boxed{288\pi}

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

We are told that a point $ P = (x, y, z) $ lies in the set $ S $ if it satisfies:
$$
d_1^2 + d_2^2 + d_3^2 = 36,
$$
where $ d_1, d_2, d_3 $ are the distances from $ P $ to the three planes:

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

---

### Step 1: Apply the Distance Formula

The distance from a point $ (x, y, z) $ to a plane $ ax + by + cz = 0 $ is given by:
$$
\text{Distance} = \frac{|ax + by + cz|}{\sqrt{a^2 + b^2 + c^2}}.
$$

So for each plane, we compute:

- **Plane 1:** $ x - z = 0 \Rightarrow a = 1, b = 0, c = -1 $
  $$
  d_1 = \frac{|x - z|}{\sqrt{1^2 + 0^2 + (-1)^2}} = \frac{|x - z|}{\sqrt{2}} \Rightarrow d_1^2 = \frac{(x - z)^2}{2}
  $$

- **Plane 2:** $ x - 2y + z = 0 \Rightarrow a = 1, b = -2, c = 1 $
  $$
  d_2 = \frac{|x - 2y + z|}{\sqrt{1^2 + (-2)^2 + 1^2}} = \frac{|x - 2y + z|}{\sqrt{6}} \Rightarrow d_2^2 = \frac{(x - 2y + z)^2}{6}
  $$

- **Plane 3:** $ x + y + z = 0 \Rightarrow a = 1, b = 1, c = 1 $
  $$
  d_3 = \frac{|x + y + z|}{\sqrt{1^2 + 1^2 + 1^2}} = \frac{|x + y + z|}{\sqrt{3}} \Rightarrow d_3^2 = \frac{(x + y + z)^2}{3}
  $$

---

### Step 2: Plug into the Given Equation

Substitute into:
$$
d_1^2 + d_2^2 + d_3^2 = 36
$$
$$
\Rightarrow \frac{(x - z)^2}{2} + \frac{(x - 2y + z)^2}{6} + \frac{(x + y + z)^2}{3} = 36
$$

Multiply both sides by 6 to eliminate denominators:
$$
3(x - z)^2 + (x - 2y + z)^2 + 2(x + y + z)^2 = 216
$$

---

### Step 3: Expand and Simplify

- First term: $ 3(x - z)^2 = 3(x^2 - 2xz + z^2) = 3x^2 - 6xz + 3z^2 $

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

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

Add these together:

- $ 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 $

- Cross terms: all cancel out (e.g., $ -6xz + 2xz + 4xz = 0 $, similar for others)

Therefore, the full equation simplifies to:
$$
6x^2 + 6y^2 + 6z^2 = 216
$$
$$
\Rightarrow x^2 + y^2 + z^2 = 36
$$

---

### Step 4: Interpret the Result

The equation $ x^2 + y^2 + z^2 = 36 $ represents a **sphere of radius 6** centered at the origin.

Set $ S $ is the surface of this sphere, and the region **enclosed** by $ S $ is the **interior** of the sphere, also known as a **ball** of radius 6 centered at the origin.

---

### Step 5: Compute the Volume of the Enclosed Region

The volume $ V $ of a sphere of radius $ r $ is:
$$
V = \frac{4}{3} \pi r^3
$$
So for $ r = 6 $:
$$
V = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi \cdot 216 = 288\pi
$$

---

### 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 equals 36. The planes are x - z = 0, x - 2y + z = 0, and x + y + z = 0.

First, let me 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 wait, the given planes are already in the form ax + by + cz = 0. So for each plane, the distance formula simplifies. Let me write down each distance squared.

Let me denote the three planes as follows:

Plane 1: x - z = 0. So, coefficients are a1 = 1, b1 = 0, c1 = -1, d1 = 0. Wait, but the general formula is ax + by + cz + d = 0, so for plane 1, it's x - z = 0, so d is 0. Therefore, the distance from P = (x, y, z) to plane 1 is |x - z| / sqrt(1² + 0² + (-1)²) = |x - z| / sqrt(2). Therefore, d1² = (x - z)² / 2.

Similarly for the other planes. Let me do the same for plane 2: x - 2y + z = 0. So coefficients a2 = 1, b2 = -2, c2 = 1, d2 = 0. Therefore, distance d2 = |x - 2y + z| / sqrt(1 + 4 + 1) = |x - 2y + z| / sqrt(6). Thus, d2² = (x - 2y + z)² / 6.

Plane 3: x + y + z = 0. Coefficients a3 = 1, b3 = 1, c3 = 1, d3 = 0. Therefore, distance d3 = |x + y + z| / sqrt(1 + 1 + 1) = |x + y + z| / sqrt(3). Hence, d3² = (x + y + z)² / 3.

So the equation given is d1² + d2² + d3² = 36. Therefore, substituting the expressions above:

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

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

Let me note that denominators are 2, 6, 3. To combine them, it might be easier to find a common denominator. Let me multiply each term by 6 to eliminate denominators:

Multiply entire equation by 6:

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

So now the equation becomes:

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 take them one by one.

First term: 3(x - z)². Let's expand (x - z)²: x² - 2xz + z². Multiply by 3: 3x² - 6xz + 3z².

Second term: (x - 2y + z)². Let me expand that. Let me denote it as [(x + z) - 2y]^2. Which is (x + z)² - 4y(x + z) + 4y². Let me compute that:

(x + z)^2 = x² + 2xz + z². Then -4y(x + z) = -4xy -4yz. Then +4y². So altogether:

x² + 2xz + z² - 4xy - 4yz + 4y².

Third term: 2(x + y + z)^2. First expand (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 + 2xz + z² - 4xy - 4yz + 4y²

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

Adding all together:

Let me compute term by term.

For x²: 3x² (from first term) + x² (second term) + 2x² (third term) = 3 + 1 + 2 = 6x².

For y²: 0 (from first term) + 4y² (second term) + 2y² (third term) = 6y².

For z²: 3z² (from first term) + z² (second term) + 2z² (third term) = 6z².

Now cross terms:

For xy: -4xy (from second term) + 4xy (from third term) = 0.

For xz: -6xz (from first term) + 2xz (from second term) + 4xz (from third term) = (-6 + 2 + 4)xz = 0xz.

For yz: -4yz (from second term) + 4yz (from third term) = 0.

Wait, so all cross terms cancel out? That leaves only the squared terms:

6x² + 6y² + 6z² = 216.

So that simplifies to 6(x² + y² + z²) = 216, divide both sides by 6:

x² + y² + z² = 36.

Wait, so the equation reduces to x² + y² + z² = 36. That is, a sphere with radius 6 centered at the origin?

But let me check my steps again because that seems surprisingly simple. Let me verify the expansion again step by step.

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

Second term: (x - 2y + z)^2. Let me expand it as (x + z - 2y)^2. Let me compute (a - b)^2 where a = x + z and b = 2y. Then it's a² - 2ab + b². So (x + z)^2 - 4y(x + z) + 4y². Which is x² + 2xz + z² - 4xy - 4yz + 4y². 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 together:

First term: 3x² -6xz +3z²

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

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

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

y² terms: 4y² (from second term) + 2y² (from third term) = 6y². Correct.

z² terms: 3z² (from first term) + z² (from second term) + 2z² (from third term) = 6z². Correct.

xy terms: -4xy (from second term) +4xy (from third term) = 0. Correct.

xz terms: -6xz (from first term) + 2xz (from second term) +4xz (from third term) = (-6 + 2 + 4)xz = 0. Correct.

yz terms: -4yz (from second term) +4yz (from third term) = 0. Correct.

So indeed, all cross terms cancel out, leaving 6x² + 6y² + 6z² = 216 => x² + y² + z² = 36.

Therefore, the equation d1² + d2² + d3² = 36 simplifies to the sphere of radius 6 centered at the origin. Therefore, the set S is a sphere with radius 6, so the volume enclosed by S is the volume of this sphere.

Wait, but the problem says "Find the region of the volume enclosed by S." So since S is a sphere, the volume enclosed is just the volume of the sphere with radius 6.

But let me think again. Wait, is there something wrong here? Because the original question mentions distances to three planes, and then through simplification, we get a sphere. However, maybe there is a mistake here? Let me check with a specific point.

For example, take the origin (0, 0, 0). Let me compute d1² + d2² + d3².

For the origin:

d1 is distance to x - z = 0: |0 - 0| / sqrt(2) = 0. So d1² = 0.

Similarly, d2 is |0 - 0 + 0| / sqrt(6) = 0, d2² = 0.

d3 is |0 + 0 + 0| / sqrt(3) = 0. So sum is 0, which is less than 36. So origin is inside S.

Now take a point on the sphere x² + y² + z² = 36. Let me take (6, 0, 0). Compute d1² + d2² + d3².

For point (6, 0, 0):

d1² = (6 - 0)^2 / 2 = 36 / 2 = 18.

d2² = (6 - 0 + 0)^2 / 6 = 36 / 6 = 6.

d3² = (6 + 0 + 0)^2 / 3 = 36 / 3 = 12.

Sum: 18 + 6 + 12 = 36. Correct. So the point (6, 0, 0) is on S. Similarly, any point on x² + y² + z² = 36 will satisfy the equation. Therefore, our derivation seems correct.

Hence, the set S is indeed the sphere of radius 6 centered at the origin, so the volume enclosed by S is (4/3)πr³ = (4/3)π*(6)^3 = (4/3)π*216 = 288π.

But the problem says "Find the region of the volume enclosed by S." So the answer is the volume of the sphere, which is 288π. However, let me check if there's some detail I missed. For example, maybe the distances are signed? But no, the distance is always non-negative, so squared distances are positive. So the equation is just the sphere.

Alternatively, maybe there's some constraint? But no, the algebra shows that the equation reduces to the sphere. Hence, the region is a sphere with radius 6, and its volume is 288π.

Wait, but let me check with another point. Let me take (0, 0, 6). Then:

d1² = (0 - 6)^2 / 2 = 36/2 = 18.

d2² = (0 - 0 + 6)^2 /6 = 36/6 = 6.

d3² = (0 + 0 + 6)^2 / 3 = 36/3 = 12.

Sum is 18 + 6 + 12 = 36. Correct. Similarly for (0, 6, 0):

d1² = (0 - 0)^2 /2 = 0.

d2² = (0 - 2*6 + 0)^2 /6 = (-12)^2 /6 = 144 /6 = 24.

d3² = (0 + 6 + 0)^2 /3 = 36 /3 = 12. Sum is 0 + 24 + 12 = 36. Correct. So yes, it works.

Therefore, the conclusion is that the set S is a sphere of radius 6, so the volume enclosed is 4/3 π * 6³ = 288 π. Therefore, the answer should be \boxed{288\pi}.

Wait, but let me check once again. The problem says "the region of the volume enclosed by S." So S is the set of points P such that d1² + d2² + d3² = 36. Which we found is the sphere x² + y² + z² = 36. Therefore, the volume inside S would be the volume of the sphere. However, is there a possibility that S is not the entire sphere? Because sometimes equations can have constraints.

But in our case, we started with d1² + d2² + d3² = 36, which translated to x² + y² + z² = 36. So S is the sphere itself, so the region enclosed by S is the interior of the sphere. But wait, the problem says "the set of points P such that d1² + d2² + d3² = 36", which is the sphere itself. However, the volume enclosed by S would be the set of points inside the sphere. Wait, but actually, the wording is a bit ambiguous. Sometimes, when they say "the region of the volume enclosed by S", they might mean the set S itself. But since S is a sphere, which is a surface, not a volume. However, the problem says "the set of points P such that d1² + d2² + d3² = 36", which is the sphere, but the question says "the region of the volume enclosed by S." So that suggests that S is the boundary of the region, which is a sphere, so the volume enclosed is the interior of the sphere.

But then, in that case, the answer is the volume of the sphere with radius 6, which is 288π. However, let me check again.

Alternatively, maybe there is some mistake here. Let me think again. Because the equation we derived is d1² + d2² + d3² = 36, which is equivalent to x² + y² + z² = 36. Therefore, the set S is the sphere of radius 6. Therefore, the volume enclosed by S would be the set of points inside or on the sphere. However, the problem says "the region of the volume enclosed by S", so if S is the sphere, then the volume enclosed is the ball of radius 6. Therefore, the answer is 4/3 π (6)^3 = 288 π.

Alternatively, maybe there's a misunderstanding here. For example, maybe the problem is not in 3D? But no, it's talking about points P in 3D space, with three planes. So it's definitely 3D.

Alternatively, perhaps I made an error in translating the distances. Let me recheck the distance formulas.

For plane x - z = 0, the distance from (x,y,z) is |x - z| / sqrt(1^2 + 0^2 + (-1)^2) = |x - z| / sqrt(2). Squared is (x - z)^2 / 2. Correct. Then same for other planes.

Then when we expanded the equation, all cross terms canceled out, leading to x² + y² + z² = 36. Therefore, the conclusion is correct.

Alternatively, maybe the problem is in another coordinate system or something? But no.

Therefore, I think the answer is 288π. Therefore, the volume enclosed by S is 288π. Therefore, the answer is \boxed{288\pi}.

**Final Answer**
\boxed{288\pi}

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

We are told that a point $ P = (x, y, z) $ lies in the set $ S $ if it satisfies:
$$
d_1^2 + d_2^2 + d_3^2 = 36,
$$
where $ d_1, d_2, d_3 $ are the distances from $ P $ to the three planes:

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

---

### Step 1: Apply the Distance Formula

The distance from a point $ (x, y, z) $ to a plane $ ax + by + cz = 0 $ is given by:
$$
\text{Distance} = \frac{|ax + by + cz|}{\sqrt{a^2 + b^2 + c^2}}.
$$

So for each plane, we compute:

- **Plane 1:** $ x - z = 0 \Rightarrow a = 1, b = 0, c = -1 $
  $$
  d_1 = \frac{|x - z|}{\sqrt{1^2 + 0^2 + (-1)^2}} = \frac{|x - z|}{\sqrt{2}} \Rightarrow d_1^2 = \frac{(x - z)^2}{2}
  $$

- **Plane 2:** $ x - 2y + z = 0 \Rightarrow a = 1, b = -2, c = 1 $
  $$
  d_2 = \frac{|x - 2y + z|}{\sqrt{1^2 + (-2)^2 + 1^2}} = \frac{|x - 2y + z|}{\sqrt{6}} \Rightarrow d_2^2 = \frac{(x - 2y + z)^2}{6}
  $$

- **Plane 3:** $ x + y + z = 0 \Rightarrow a = 1, b = 1, c = 1 $
  $$
  d_3 = \frac{|x + y + z|}{\sqrt{1^2 + 1^2 + 1^2}} = \frac{|x + y + z|}{\sqrt{3}} \Rightarrow d_3^2 = \frac{(x + y + z)^2}{3}
  $$

---

### Step 2: Plug into the Given Equation

Substitute into:
$$
d_1^2 + d_2^2 + d_3^2 = 36
$$
$$
\Rightarrow \frac{(x - z)^2}{2} + \frac{(x - 2y + z)^2}{6} + \frac{(x + y + z)^2}{3} = 36
$$

Multiply both sides by 6 to eliminate denominators:
$$
3(x - z)^2 + (x - 2y + z)^2 + 2(x + y + z)^2 = 216
$$

---

### Step 3: Expand and Simplify

- First term: $ 3(x - z)^2 = 3(x^2 - 2xz + z^2) = 3x^2 - 6xz + 3z^2 $

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

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

Add these together:

- $ 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 $

- Cross terms: all cancel out (e.g., $ -6xz + 2xz + 4xz = 0 $, similar for others)

Therefore, the full equation simplifies to:
$$
6x^2 + 6y^2 + 6z^2 = 216
$$
$$
\Rightarrow x^2 + y^2 + z^2 = 36
$$

---

### Step 4: Interpret the Result

The equation $ x^2 + y^2 + z^2 = 36 $ represents a **sphere of radius 6** centered at the origin.

Set $ S $ is the surface of this sphere, and the region **enclosed** by $ S $ is the **interior** of the sphere, also known as a **ball** of radius 6 centered at the origin.

---

### Step 5: Compute the Volume of the Enclosed Region

The volume $ V $ of a sphere of radius $ r $ is:
$$
V = \frac{4}{3} \pi r^3
$$
So for $ r = 6 $:
$$
V = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi \cdot 216 = 288\pi
$$

---

### 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, let's see. Yann and Camille are going to a restaurant with 10 items on the menu. Each of them orders one dish, and they don't want to order the same dish. The question is asking how many different combinations of meals they can order. Also, it mentions that it matters who orders what, so Yann ordering chicken and Camille ordering fish is different from Yann ordering fish and Camille ordering chicken.

Alright, so first, I need to figure out the total number of possible combinations without any restrictions and then subtract the ones where they order the same dish? Or maybe there's a more straightforward way. Let me think.

Since each person is ordering a dish, and there are 10 dishes, for Yann, he has 10 choices. Then, once Yann has ordered a dish, Camille can't order the same one. So Camille would have 9 choices left. So maybe it's 10 multiplied by 9? Let me check.

Alternatively, if I think of it as permutations. Since the order matters (because who orders what matters), it's like arranging two different dishes where the first one is Yann's and the second is Camille's. So the number of permutations of 10 dishes taken 2 at a time. Which is 10 * 9 = 90. That seems right.

Wait, let me make sure I'm not missing anything. Let's think of it step by step. For Yann, there are 10 options. For each of Yann's choices, Camille can't choose that same dish. So, if Yann picks dish A, Camille has 9 dishes left. So for each of Yann's 10 choices, there are 9 options for Camille. Therefore, total combinations are 10 * 9 = 90. Yeah, that makes sense.

Alternatively, if I think of the total number of possible combinations without restrictions, it would be 10 * 10 = 100, since each can choose any dish. Then subtract the cases where they order the same dish. How many such cases are there? Well, if they both order the same dish, there are 10 possibilities (both order dish 1, both order dish 2, etc.). So total valid combinations would be 100 - 10 = 90. Same answer. So that confirms it.

Therefore, the answer should be 90. Let me think again if there's another way to approach this.

Another way: For the first person, Yann, there are 10 choices. For Camille, since she can't choose the same as Yann, there are 9 choices. So total ways are 10 * 9 = 90. Yep, same result.

Alternatively, using combinations. Wait, but combinations don't consider order. If we thought of it as choosing two different dishes and then assigning them to Yann and Camille. So first, how many ways to choose two different dishes? That would be C(10, 2) = 45. Then, for each pair of dishes, there are 2 ways to assign them to Yann and Camille (Yann gets dish A and Camille dish B, or vice versa). So total is 45 * 2 = 90. Yep, same answer again.

So regardless of the method, it seems like 90 is the answer. Therefore, I think that's the correct answer.

**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 key points:

- There are **10 items on the menu**.
- Each person orders **one dish**.
- **They refuse to order the same dish**.
- **The order matters**: Yann ordering chicken and Camille ordering fish is considered different from Yann ordering fish and Camille ordering chicken.

---

### Step-by-Step Reasoning:

1. **Yann's choices**:
   Yann can order any of the 10 dishes on the menu.
   So, Yann has **10 choices**.

2. **Camille's choices (after Yann)**:
   Since Camille **cannot order the same dish** as Yann, and there are no other restrictions, she now has **9 remaining choices**.

3. **Total combinations**:
   For **each of the 10 choices Yann can make**, Camille will have 9 distinct valid meals.
   Therefore, the total number of combinations is:
   $$
   10 \times 9 = 90
   $$

---

### Alternative Method (Verification via Combinations and Permutations):

- **Total number of unrestricted combinations**:
  If there were no restriction, Yann and Camille could each choose any dish, resulting in:
  $$
  10 \times 10 = 100
  $$

- **Subtract the invalid combinations**:
  There are 10 combinations where both Yann and Camille order the **same dish** (e.g., both choose dish 1, both choose dish 2, ..., both choose dish 10).
  So, subtract these:
  $$
  100 - 10 = 90
  $$

---

### Using Permutations:

This is a **permutation** problem: choosing 2 different dishes (in order) from 10.
The number of permutations of 10 items taken 2 at a time is:
$$
P(10, 2) = 10 \times 9 = 90
$$

---

### Final Answer:

$$
\boxed{90}
$$

[Assistant]
Explanation:
Okay, let's see. Yann and Camille are going to a restaurant with 10 items on the menu. Each of them orders one dish, and they don't want to order the same dish. The question is asking how many different combinations of meals they can order. Also, it mentions that it matters who orders what, so Yann ordering chicken and Camille ordering fish is different from Yann ordering fish and Camille ordering chicken.

Alright, so first, I need to figure out the total number of possible combinations without any restrictions and then subtract the ones where they order the same dish? Or maybe there's a more straightforward way. Let me think.

Since each person is ordering a dish, and there are 10 dishes, for Yann, he has 10 choices. Then, once Yann has ordered a dish, Camille can't order the same one. So Camille would have 9 choices left. So maybe it's 10 multiplied by 9? Let me check.

Alternatively, if I think of it as permutations. Since the order matters (because who orders what matters), it's like arranging two different dishes where the first one is Yann's and the second is Camille's. So the number of permutations of 10 dishes taken 2 at a time. Which is 10 * 9 = 90. That seems right.

Wait, let me make sure I'm not missing anything. Let's think of it step by step. For Yann, there are 10 options. For each of Yann's choices, Camille can't choose that same dish. So, if Yann picks dish A, Camille has 9 dishes left. So for each of Yann's 10 choices, there are 9 options for Camille. Therefore, total combinations are 10 * 9 = 90. Yeah, that makes sense.

Alternatively, if I think of the total number of possible combinations without restrictions, it would be 10 * 10 = 100, since each can choose any dish. Then subtract the cases where they order the same dish. How many such cases are there? Well, if they both order the same dish, there are 10 possibilities (both order dish 1, both order dish 2, etc.). So total valid combinations would be 100 - 10 = 90. Same answer. So that confirms it.

Therefore, the answer should be 90. Let me think again if there's another way to approach this.

Another way: For the first person, Yann, there are 10 choices. For Camille, since she can't choose the same as Yann, there are 9 choices. So total ways are 10 * 9 = 90. Yep, same result.

Alternatively, using combinations. Wait, but combinations don't consider order. If we thought of it as choosing two different dishes and then assigning them to Yann and Camille. So first, how many ways to choose two different dishes? That would be C(10, 2) = 45. Then, for each pair of dishes, there are 2 ways to assign them to Yann and Camille (Yann gets dish A and Camille dish B, or vice versa). So total is 45 * 2 = 90. Yep, same answer again.

So regardless of the method, it seems like 90 is the answer. Therefore, I think that's the correct answer.

**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 key points:

- There are **10 items on the menu**.
- Each person orders **one dish**.
- **They refuse to order the same dish**.
- **The order matters**: Yann ordering chicken and Camille ordering fish is considered different from Yann ordering fish and Camille ordering chicken.

---

### Step-by-Step Reasoning:

1. **Yann's choices**:
   Yann can order any of the 10 dishes on the menu.
   So, Yann has **10 choices**.

2. **Camille's choices (after Yann)**:
   Since Camille **cannot order the same dish** as Yann, and there are no other restrictions, she now has **9 remaining choices**.

3. **Total combinations**:
   For **each of the 10 choices Yann can make**, Camille will have 9 distinct valid meals.
   Therefore, the total number of combinations is:
   $$
   10 \times 9 = 90
   $$

---

### Alternative Method (Verification via Combinations and Permutations):

- **Total number of unrestricted combinations**:
  If there were no restriction, Yann and Camille could each choose any dish, resulting in:
  $$
  10 \times 10 = 100
  $$

- **Subtract the invalid combinations**:
  There are 10 combinations where both Yann and Camille order the **same dish** (e.g., both choose dish 1, both choose dish 2, ..., both choose dish 10).
  So, subtract these:
  $$
  100 - 10 = 90
  $$

---

### Using Permutations:

This is a **permutation** problem: choosing 2 different dishes (in order) from 10.
The number of permutations of 10 items taken 2 at a time is:
$$
P(10, 2) = 10 \times 9 = 90
$$

---

### Final Answer:

$$
\boxed{90}
$$

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.