Question

A committee of four is formed from five eligible members. Let the eligible members be represented with A, B, C, D, and E. The possible outcomes include S = {ABCD, BCDE, ACDE, ABCE, ABDE}. Which statements about the situation are true? Check all that apply.

There are 120 different ways to choose the committee.
If person A must be on the committee, there is only one way to form the committee.
If persons A and C must be on the committee, there are three ways to form the committee.
There are five ways to form the committee if person E must be on it.
If the number of eligible members increases, the number of outcomes increases.

Answer 1

True statements:
- If persons A and C must be on the committee, there are three ways to form the committee.
- If the number of eligible members increases, the number of outcomes increases.

Explanations:
- Total ways to choose 4 from 5 is C(5,4)=5, not 120 (120 = 5!).
- If A must be on the committee, choose the other 3 from the remaining 4: C(4,3)=4 ways (not 1).
- If A and C must be on the committee, choose the other 2 from the remaining 3: C(3,2)=3 ways.
- If E must be on the committee, choose the other 3 from the remaining 4: C(4,3)=4 ways (not 5).
- For fixed committee size, C(n,4) increases as n (eligible members) increases, so more eligible members → more outcomes.