<Question>
Yann and Camille go to a restaurant.  If there are 10 items on the menu, and each orders one dish, how many different combinations of meals can Yann and Camille order if they refuse to order the same dish? (It does matter who orders what---Yann ordering chicken and Camille ordering fish is different from Yann ordering fish and Camille ordering chicken.)
</Question>

<think>
<Plan_1>
Okay, so Yann and Camille are going to a restaurant where there are 10 items on the menu. Each of them orders one dish, and they don't want to order the same dish. The problem is asking how many different combinations of meals they can order, considering that it matters who orders what. For example, Yann ordering chicken and Camille ordering fish is different from Yann ordering fish and Camille ordering chicken. 

Let me start by understanding the problem step by step. First, there are 10 dishes on the menu. Yann picks one, then Camille picks one. But they can't pick the same one. Also, the order matters because who is ordering which dish matters. So, if Yann has 10 choices, then Camille would have 9 choices since she can't pick the same as Yann. So maybe it's 10 multiplied by 9? That would be 90. But let me make sure I'm not missing something here.

Wait, but maybe I should think of it as permutations.

<prompt_1.1> Consider permutations and confirm if the initial calculation of 10 × 9 = 90 is correct. </prompt_1.1>

But let me check with a smaller number to see if this makes sense. Suppose there are 2 dishes on the menu: Dish A and Dish B. If Yann and Camille each order one, and they can't order the same dish. How many combinations are there?

<prompt_1.2> Use a smaller example with 2 dishes to validate the calculation and confirm the result. </prompt_1.2>

Another way to think about it: For each of Yann's choices, there are 10 possibilities. Then, for each of those, Camille has 9. So, the total number is 10*9=90.

<prompt_1.3> Explore alternative ways to confirm the calculation and ensure the logic is sound. </prompt_1.3>

Alternatively, if the problem had said that the dishes are indistinct, meaning that the order doesn't matter, then it would be combinations, which is 10 choose 2, which is 45. But the problem here says that it matters who orders what, so the order is important.
Therefore, permutations, not combinations.
So 10*9=90.

Let me check once again with another example. Suppose there are 3 dishes: A, B, C.

<prompt_1.4> Use another example with 3 dishes to validate the calculation and confirm the result. </prompt_1.4>
</Plan_1>

<Execution_1>
<execution_1.1>
Permutations are arrangements where order matters, combinations where order doesn't. Since in this case, the problem specifies that it matters who orders what, so the order here is important. So, if Yann has 10 choices, Camille has 9, so 10 * 9 = 90. That seems right. But let me make sure I'm not confusing permutations with combinations here. 

Alternatively, if the problem had said that they are just choosing two different dishes regardless of who orders which, then it would be combinations, but since the problem states that it matters who orders what, then each different assignment is unique. So, for example, Yann choosing A and Camille choosing B is different from Yann choosing B and Camille choosing A.
</execution_1.1>

<execution_1.2>
If Yann picks Dish A, Camille has to pick Dish B. If Yann picks Dish B, Camille has to pick Dish A. So that's 2 possible combinations. Which is 2*1=2. So yes, that works. So in the case of 10 dishes, it should be 10*9=90.
</execution_1.2>

<execution_1.3>
Let's list them out. If there are 2 dishes, then Yann can choose A or B. If Yann chooses A, Camille has to choose B. If Yann chooses B, Camille has to choose A. So that's 2 possibilities. Wait, but according to the formula, it would be 2*1=2, which matches. So that works. 

Another example: 3 dishes. If Yann picks first, then Camille has 2 choices. So for each of Yann's 3 choices, Camille has 2. So 3*2=6. Let's list them: If dishes are A, B, C. Yann can have A, B, or C. If Yann has A, Camille can have B or C. If Yann has B, Camille can have A or C. If Yann has C, Camille can have A or B. So that's 3*2=6 combinations. Which makes sense. So in the case of 10 dishes, it's 10*9=90.
</execution_1.3>

<execution_1.4>
If Yann picks A, Camille can pick B or C. If Yann picks B, Camille can pick A or C. If Yann picks C, Camille can pick A or B. So for each of Yann's 3 choices, Camille has 2. Total is 3*2=6. Which is 3*2=6. Which is 6. If we list them: AB, BA, AC, CA, BC, CB. That's 6, which is 3*2. So yes, in that case, it's 6. So the same logic would apply for 10 dishes. So 10*9=90. Therefore, the answer should be 90.
</execution_1.4>
</Execution_1>

<Plan_2>
Based on execution_1.1, execution_1.2, execution_1.3, execution_1.4:
Therefore, the answer should be 10*9=90. 

Wait, but let me make sure there isn't any trick here.

<prompt_2.1> Provide final reasoning and conclude the solution to the problem, confirming the final answer. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
Like, is there any possibility that the problem is considering that they can't order the same dish, but since they are different people, does that affect anything? No, because even if they are the same person, but here they are different people. So, if Yann orders a dish, Camille just can't order the same one. So, the calculation is 10 choices for Yann, then 9 for Camille. So 10*9=90. 

Alternatively, if you think of it as permutations of 10 dishes taken 2 at a time, the formula is P(n, k) = n! / (n - k)! , where n=10, k=2. So, 10! / (10-2)! = 10! / 8! = 10*9*8! / 8! = 10*9=90. So that also gives 90. 

Therefore, I think the answer is 90. I don't see any other factors here. The key points are that each person orders one dish, they can't order the same one, and the order matters because it's specified who is ordering which. So, no overlap or overcounting. Each choice is independent once the previous person's choice is known. So, 10*9=90.
</execution_2.1>
</Execution_2>

<Final_answer>
Therefore, the number of different combinations of meals Yann and Camille can order, under the given conditions, is $\boxed{90}$.
</Final_answer>
</think>