In how many ways can a committee consisting of 3 men and 2 women be formed from 7 men and 5 women?

Options

A. A) 45
B. B) 350
C. C) 700
D. D) 4200

Answer

To solve the problem of forming a committee consisting of 3 men and 2 women from a group of 7 men and 5 women, we can use the concept of combinations.

### Step-by-Step Solution:

1. **Identify the Groups**:
- We need to select 3 men from a total of 7 men.
- We also need to select 2 women from a total of 5 women.

2. **Use the Combination Formula**:
- The formula for combinations is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
- Where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

3. **Calculate the Number of Ways to Choose Men**:
- For selecting 3 men from 7:
\[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7!}{3! \cdot 4!} \]
- Simplifying this:
\[ \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = \frac{210}{6} = 35 \]

4. **Calculate the Number of Ways to Choose Women**:
- For selecting 2 women from 5:
\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \cdot 3!} \]
- Simplifying this:
\[ \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = \frac{20}{2} = 10 \]

5. **Combine the Results**:
- Since the selections of men and women are independent, we multiply the number of ways to choose the men by the number of ways to choose the women:
\[ \text{Total Ways} = \binom{7}{3} \times \binom{5}{2} = 35 \times 10 = 350 \]

### Final Answer:
The total number of ways to form a committee consisting of 3 men and 2 women from 7 men and 5 women is **350**.

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