Question

Out of 5 men and 2 women, a committee of 3 is to be formed. In how many ways can it be formed if at least one woman is included in each committee?

A. 21
B. 25
C. 32
D. 50

Answer 1

The correct answer is B. 25. To form a committee of 3 members with at least one woman from a group of 5 men and 2 women, subtract the number of all-male committees from the total possible committees. Using combinations, the answer is 25 different committees.

Step-by-step explanation:

To answer the question, we need to calculate the number of ways to form a committee of 3 members from a group of 5 men and 2 women with the condition that at least one woman is on each committee. Here's a step-by-step explanation using the concept of combinations:

• First, calculate the total number of committees that include at least one woman, which can be done by subtracting the number of all-male committees from the total number of possible committees.
• The number of ways to choose 3 men from 5 is calculated as a combination: C(5, 3).
• The total number of ways to choose a committee of 3 from 7 people is C(7, 3).
• To find the number of committees with at least one woman, subtract the all-male committees from the total: C(7, 3) - C(5, 3).

Using combination formulas, we find C(5, 3) = 10 and C(7, 3) = 35. Thus, the number of committees with at least one woman is 35 - 10 = 25 committees. The correct answer is B. 25.