A committee of 6 is to be chosen from the 25 members of the student council (including the
executive) to organize Spirit Week. In how many ways can this be done if:
a) there are no restrictions
b) there are 14 junior members and 11 senior members; the committee must have 4 juniors and 2 seniors
a) To find the number of ways to choose a committee of 6 without any restrictions, you can use the formula for combinations. The number of ways to choose a committee of 6 out of 25 members can be calculated as:
C(25, 6) = 25! / (6! * (25-6)!)
where "C" denotes the combination, "!" denotes factorial, and (25-6) is used to represent the remaining members who are not chosen.
Using this formula, we can solve:
C(25, 6) = 25! / (6! * 19!)
Note that 6! (factorial of 6) represents the number of ways to arrange the chosen committee members among themselves, as the order of the committee members does not matter. Also, 19! (factorial of 19) represents the remaining members who are not chosen, as the committee size is fixed at 6.
Calculating this expression will give you the final answer.
b) In this case, you need to choose a committee of 6 with 4 juniors and 2 seniors from a group of 14 juniors and 11 seniors.
To determine the number of ways to form this committee, you need to calculate the product of two combination values: one for selecting 4 juniors from 14 and another for selecting 2 seniors from 11.
Combining these combinations gives:
C(14, 4) * C(11, 2)
Where C(14, 4) represents the number of ways to choose 4 juniors out of 14, and C(11, 2) represents the number of ways to choose 2 seniors out of 11.
Calculating this expression will give you the final answer.