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

Options

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

Explanation

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 follow these steps:

### Step 1: Determine the number of ways to choose 3 men from 7 men.
We can use the combination formula, which is given by:

\[ nCr = \frac{n!}{r!(n-r)!} \]

In this case, \( n = 7 \) and \( r = 3 \):

\[ 7C3 = \frac{7!}{3!(7-3)!} = \frac{7!}{3! \cdot 4!} \]

### Step 2: Calculate \( 7C3 \).
Breaking it down:

\[ 7C3 = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = \frac{210}{6} = 35 \]

### Step 3: Determine the number of ways to choose 2 women from 5 women.
Using the combination formula again, where \( n = 5 \) and \( r = 2 \):

\[ 5C2 = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \cdot 3!} \]

### Step 4: Calculate \( 5C2 \).
Breaking it down:

\[ 5C2 = \frac{5 \times 4}{2 \times 1} = \frac{20}{2} = 10 \]

### Step 5: Calculate the total number of ways to form the committee.
To find the total number of ways to form the committee, we multiply the number of ways to choose the men by the number of ways to choose the women:

\[ \text{Total ways} = 7C3 \times 5C2 = 35 \times 10 = 350 \]

### 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**.

---