Question

Out of 5 men and 3 women is to be formed. Omen, a committee of 4 members is to be formed. In how many ways can it be done if the committee includes at least one woman ? A. 20 b. 35 c. 30 d. 15

Answer 1

Answer: 65

Step-by-step explanation:

Let us name each of the men A, B, C, D and E respectively

Let us name each of the women X, Y and Z respectively

Since the condition requires at least one woman on the committee, let us take 3 cases into consideration:

1) When only 1 woman is on the committee:

• There are 3 men on the committee, regardless of which women is included. Thus, we need to find the number of unordered combinations of 3 men we can obtain from the given 5 men. These are: ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE (10 combinations)
• Each woman will be paired with either of these combinations making the total number of combinations where only 1 woman is present 3x10=30

2)When only 2 women are on the committee:

• There are two men on the committee, the number of unordered combinations of men and women both need to be considered.
• For women, combinations are: XY, YZ, XZ (3 combinations)
• For men, combinations are: AB, AC, AD, AE, BC, BD, BE, CD, CE, DE (10 combinations)
• Total number of possibilities: 3x10=30

3) When all 3 women are in the committee

• All 3 women are included and thus there is only one unordered combinations (XYZ)
• Only one man is on the committee and thus possible teams will include either of the 5 men, number of combinations of men will be 5
• This makes number of possibilities 1x5=5

When we add all of the outcomes of different cases we get: 30+30+5= 65. Given options are wrong, correct answer is 65.