Question

this year, there are ten freshmen, nine sophomores, seven juniors, and ten seniors are eligible to be on a committee. in how many ways can a dance committee of 8 students be chosen?

Answer 1

To find the number of ways a dance committee of 8 students can be chosen from four groups (freshmen, sophomores, juniors, and seniors), we need to use combinations. The formula for the number of ways is C(10, x) × C(9, y) × C(7, z) × (8 - x - y - z).

Step-by-step explanation:

To find the number of ways a dance committee of 8 students can be chosen, we need to use the concept of combinations. We have four groups of students: freshmen, sophomores, juniors, and seniors. From each group, we need to select a certain number of students to be on the committee. Let's break it down into steps:

1. Choose the number of freshmen on the committee. There are 10 freshmen, so we can choose any number from 0 to 8. This can be done using combinations: C(10, x), where x is the number of freshmen chosen.
2. Choose the number of sophomores on the committee. There are 9 sophomores, so we can choose any number from 0 to (8 - x) to ensure we fill the remaining spots. This can also be done using combinations: C(9, y), where y is the number of sophomores chosen.
3. Choose the number of juniors on the committee. There are 7 juniors, so we can choose any number from 0 to (8 - x - y) to ensure we fill the remaining spots. Again, combinations can be used: C(7, z), where z is the number of juniors chosen.
4. The number of seniors on the committee will be the remaining spots, so the number of seniors chosen will be (8 - x - y - z).
5. Finally, we need to multiply the number of possibilities from steps 1 to 4 together to get the total number of ways the committee can be chosen.

Therefore, the formula for the number of ways a dance committee of 8 students can be chosen is: C(10, x) × C(9, y) × C(7, z) × (8 - x - y - z).

Answer 2

The number of ways to form a dance committee of 8 students from a pool of 36 eligible students (sum of all classes) can be calculated using the combination formula C(n, k).

Step-by-step explanation:

The number of ways to choose a dance committee of 8 students from the available ten freshmen, nine sophomores, seven juniors, and ten seniors can be calculated using combinatorics. This is not a hypergeometric problem because the question does not differentiate between the classes of the students or predefine a specific number of students required from each class. Instead, it's a straightforward application of the combination formula.

First, compute the total number of eligible students, which is 10 freshmen + 9 sophomores + 7 juniors + 10 seniors = 36 students. Next, calculate the number of ways to choose 8 out of these 36 students, which is given by the combination formula C(n, k) = n! / (k!(n - k)!), where n is the total number of items, and k is the number of items to choose.

Therefore, the total number of ways to form the committee is C(36, 8), which is a calculation of 36! divided by the product of 8! and (36-8)!.