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:

Split the requirement into two separate, independent tasks: picking the men and picking the woman. Work out each count on its own using combinations, then combine the two results with the multiplication principle since both selections have to happen together for a valid committee.

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.

Answer 2

Concept:

• When people are picked one after another, the count changes if the order of picking matters. Since a committee has no order, treating a selection as ordered first, then correcting for the repeated arrangements, still gives the right count.
• Picking a single person has no ordering to correct for, since one person cannot be arranged in more than one way.

Step 1: Count the ordered ways to pick 2 men
There are 5 men. The first man can be chosen in 5 ways, and the second man can then be chosen from the remaining 4 men, giving $5 \times 4 = 20$ ordered picks.

Step 2: Remove the overcounting caused by order
Every pair of 2 men has been counted twice above, once for each order the pair could have been picked in, since 2 men can be arranged among themselves in $2! = 2$ ways. So the number of unordered pairs of men is $\frac{20}{2} = 10$.

Step 3: Count the ways to pick 1 woman
Only 1 woman is needed out of 4, so there is no order to correct for. The number of ways is simply $4$.

Step 4: Combine using the multiplication principle
Choosing the men and choosing the woman are two independent tasks that both need to happen, so the counts are multiplied: $10 \times 4 = 40$.

Final Answer: The committee can be formed in 40 ways.