Question

A committee of 3 is to be formed from 5 men and 4 women. It must contain exactly 2 men and 1 woman. How many ways can it be done?

• A. 30
• B. 50
• C. 20
• D. 40

Hint:

In combination problems, remember the keywords. "And" usually means you need to multiply the number of ways (like selecting men AND women), while "Or" usually means you need to add the ways.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We need to form a committee of 3 people with a specific composition: exactly 2 men and 1 woman. The available pool of candidates consists of 5 men and 4 women. The task is to find the total number of ways to form such a committee.
Step 2: Key Formula or Approach:
This is a problem of combinations, as the order in which the members are selected for the committee does not matter. The formula for combinations is: \[ ^nC_r = \frac{n!}{r!(n-r)!} \] where n is the total number of items to choose from, and r is the number of items to choose. The total number of ways will be the product of the number of ways to select the men and the number of ways to select the women.
Step 3: Detailed Explanation:
Part 1: Selecting the men
We need to select 2 men from a group of 5. The number of ways to do this is: \[ ^5C_2 = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = \frac{5 \times 4}{2} = 10 \] So, there are 10 ways to choose the 2 men.
Part 2: Selecting the woman
We need to select 1 woman from a group of 4. The number of ways to do this is: \[ ^4C_1 = \frac{4!}{1!(4-1)!} = \frac{4!}{1!3!} = \frac{4 \times 3!}{1 \times 3!} = 4 \] So, there are 4 ways to choose the 1 woman.
Part 3: Total number of ways
To find the total number of ways to form the committee, we multiply the number of ways of selecting men and women (using the multiplication principle of counting). \[ \text{Total ways} = (\text{Ways to select men}) \times (\text{Ways to select women}) \] \[ \text{Total ways} = 10 \times 4 = 40 \] Step 4: Final Answer:
There are 40 different ways to form the committee with exactly 2 men and 1 woman. Therefore, option (D) is the correct answer.