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

Options

A. (a) 45
B. (b) 350
C. (c) 4200
D. (d) 230

Answer

To solve the problem of how many ways a committee consisting of 3 men and 2 women can be chosen from 7 men and 5 women, we can use the concept of combinations.

### Step-by-Step Solution:

1. **Identify the total number of men and women:**
- Total men = 7
- Total women = 5

2. **Determine the number of men to choose:**
- We need to choose 3 men from the 7 available men.

3. **Use the combination formula for men:**
- The number of ways to choose r items from n items is given by the formula:
\[ \binom{n}{r} = \frac{n!}{(n-r)! \cdot r!} \]
- For choosing 3 men from 7, we can express this as:
\[ \binom{7}{3} = \frac{7!}{(7-3)! \cdot 3!} = \frac{7!}{4! \cdot 3!} \]

4. **Calculate \(\binom{7}{3}\):**
- Expanding the factorials:
\[ \binom{7}{3} = \frac{7 \times 6 \times 5 \times 4!}{4! \times 3 \times 2 \times 1} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} \]
- Simplifying:
\[ = \frac{210}{6} = 35 \]

5. **Determine the number of women to choose:**
- We need to choose 2 women from the 5 available women.

6. **Use the combination formula for women:**
- For choosing 2 women from 5, we can express this as:
\[ \binom{5}{2} = \frac{5!}{(5-2)! \cdot 2!} = \frac{5!}{3! \cdot 2!} \]

7. **Calculate \(\binom{5}{2}\):**
- Expanding the factorials:
\[ \binom{5}{2} = \frac{5 \times 4 \times 3!}{3! \times 2 \times 1} = \frac{20}{2} = 10 \]

8. **Combine the results:**
- The total number of ways to form the committee is the product of the combinations of men and women:
\[ \text{Total ways} = \binom{7}{3} \times \binom{5}{2} = 35 \times 10 = 350 \]

### Final Answer:
Thus, the total number of ways to form the committee is **350**.

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