Question

finding outcomes
pre-test complete
time remaining
59:37
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:

Step1: Verify total committee count

The sample space $S = \{ABC, ABD, ACD, BCD\}$ has 4 elements, so total ways = 4.

Step2: Check committees with D

Outcomes with D: $ABD, ACD, BCD$ → 3 ways.

Step3: Analyze 7-eligible member combinations

Number of 3-person committees from 7 is $\binom{7}{3} = \frac{7!}{3!(7-3)!} = 35$, which is more than 4, so this is false.

Step4: Check committees with B and C

Only 1 outcome with B and C: $ABC$ → not 2 ways, so this is false.

Step5: Check committees with A and C

Only 1 outcome with A and C: $ACD$ → 1 way.

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 A and C must be on the committee, then there is only one way to form the committee.