Question

A membership committee of three is formed from four eligible members. Let the eligible members be represented by A, B, C, and D. The possible outcomes include S = {ABC, ABD, ACD, BCD}.

Which statements about the situation are true? Select three options.

There are four ways to choose the committee.
There are three ways to form the committee if person D must be on it.
If seven members are eligible next year, then there will be fewer combinations.
If persons B and C must be on the committee, there are two ways to form the committee.
If persons A and C must be on the committee, then there is only one way to form the committee.

Answer 1

True statements (three):

- There are four ways to choose the committee.
- Reason: C(4,3) = 4 (ABC, ABD, ACD, BCD).

- There are three ways to form the committee if person D must be on it.
- Reason: D fixed, choose 2 of the remaining 3: C(3,2) = 3.

- If persons B and C must be on the committee, there are two ways to form the committee.
- Reason: B and C fixed, choose the third from {A,D}: 2 ways.

False statements:

- If seven members are eligible next year, there will be fewer combinations. (False: C(7,3)=35, which is more.)
- If persons A and C must be on the committee, then there is only one way to form the committee. (False: third member can be B or D, so 2 ways.)