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

Explanation:

1. For "There are four ways to choose the committee": The sample space \( S=\{ABC, ABD, ACD, BCD\} \) has 4 elements, so the number of ways is 4. This is true.
2. For "There are three ways to form the committee if person D must be on it": If D is on the committee, we need to choose 2 more from A, B, C. The combinations are ABD, ACD, BCD (3 combinations). This is true.
3. For "If seven members are eligible next year, then there will be fewer combinations": The number of combinations of \( n \) things taken \( r \) at a time is \( C(n,r)=\frac{n!}{r!(n - r)!} \). For \( n = 4,r=3 \), \( C(4,3)=4 \); for \( n = 7,r = 3 \), \( C(7,3)=\frac{7!}{3!4!}=\frac{7\times6\times5}{3\times2\times1}=35 \), which is more than 4. So this is false.
4. For "If persons B and C must be on the committee, there are two ways to form the committee": If B and C are on the committee, we need to choose 1 more from A and D. The combinations are ABC and BCD (2 combinations). This is true.
5. For "If persons A and C must be on the committee, then there is only one way to form the committee": If A and C are on the committee, we need to choose 1 more from B and D. The combinations are ABC and ACD (2 combinations), not 1. So this is false.

Answer:

• There are four ways to choose the committee.
• There are three ways to form the committee if person D must be on it.
• If persons B and C must be on the committee, there are two ways to form the committee.