A committee of 6 students is to be selected from 17 students in a fraternity.
a. In how many ways can this be done?
b. In how many ways can the group that will not take part be chosen?
a. The number of ways to form a 6 -student committee is $\square$.
Answer
The number of ways to select the group that will not take part is the same as the number of ways to select the committee, because for each committee, there is a unique group of students that is not part of the committee. Therefore, the number of ways to select the group that will not take part is also \(\boxed{12376}\).
Step 1 ：The problem is asking for the number of ways to select 6 students out of 17. This is a combination problem, because the order in which the students are selected does not matter.
Step 2 ：The formula for combinations is: \(C(n, k) = \frac{n!}{k!(n-k)!}\) where n is the total number of items, k is the number of items to choose, and '!' denotes factorial, which is the product of all positive integers up to that number. In this case, n=17 and k=6.
Step 3 ：Substituting the given values into the formula, we get: \(C(17, 6) = \frac{17!}{6!(17-6)!}\)
Step 4 ：Calculating the above expression, we find that there are 12376 different ways to select a committee of 6 students from a group of 17.
Step 5 ：Final Answer: The number of ways to form a 6 -student committee is \(\boxed{12376}\).
Step 6 ：The number of ways to select the group that will not take part is the same as the number of ways to select the committee, because for each committee, there is a unique group of students that is not part of the committee. Therefore, the number of ways to select the group that will not take part is also \(\boxed{12376}\).
