Final answer:
To solve this problem, we need to use combinations to count the different ways of selecting students from the three classes. By considering different cases, we calculate the total number of ways and find that k is equal to 17850.
Step-by-step explanation:
To solve this problem, we will use combinations. We need to select 10 students from the three classes, ensuring that at least 2 students are selected from each class and at most 5 students are selected from the combined classes of 10 and 11. Let's break down the problem into different cases:
Case 1: Selecting 2 students from class 10, 2 students from class 11, and 6 students from class 12.
The number of ways to select 2 students from class 10 is C(5, 2) = 10.
The number of ways to select 2 students from class 11 is C(6, 2) = 15.
The number of ways to select 6 students from class 12 is C(8, 6) = 28.
So, the total number of ways for Case 1 is 10 * 15 * 28 = 4200.
Case 2: Selecting 3 students from class 10, 2 students from class 11, and 5 students from class 12.
The number of ways to select 3 students from class 10 is C(5, 3) = 10.
The number of ways to select 2 students from class 11 is C(6, 2) = 15.
The number of ways to select 5 students from class 12 is C(8, 5) = 56.
So, the total number of ways for Case 2 is 10 * 15 * 56 = 8400.
Case 3: Selecting 4 students from class 10, 2 students from class 11, and 4 students from class 12.
The number of ways to select 4 students from class 10 is C(5, 4) = 5.
The number of ways to select 2 students from class 11 is C(6, 2) = 15.
The number of ways to select 4 students from class 12 is C(8, 4) = 70.
So, the total number of ways for Case 3 is 5 * 15 * 70 = 5250.
Adding up the total number of ways from each case, we get 4200 + 8400 + 5250 = 17850. Therefore, the value of k is 17850.