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.

A
The first statement is false. The total number of ways of choosing 4 members from 5 is listed in the body of the question. There are 5 ways without any restrictions.

B
is false. If A must be on the committee then there are 4 ways of making forming he committee. What happens is that B C D and E are in turn left off the committee. 

C
If A and C are both on the committee then  this statement is true. ABCD ACDE and ACBE are the three ways.

D
If E must be on it, the answer is the same as the answer for Choice B. This statement is false.

E
The question is ambiguous. How many people do you get to choose from? If you are only going to choose a committee of 4 then the answer is true. I will assume that is what is meant, but only the person making up the question can clear up the ambiguity.

The third and possibly the last statement are true.