Ajay has 7 relatives, 3 men and 4 women. His wife Kajol also has 7 relatives, 3 women and 4 men. In how many ways can they invite 3 men and 3 women so that 3 of them are the Ajay's relatives and 3 his wife Kajol's?

Options

A. 485
B. 458
C. 365
D. none of these

Answer

To solve the problem of how many ways Ajay and his wife Kajol can invite 3 men and 3 women from their relatives, we will break it down into cases based on the distribution of men and women from each side.

### Step-by-Step Solution:

1. **Understanding the relatives**:
- Ajay has 3 men and 4 women.
- Kajol has 4 men and 3 women.

2. **Total invitation requirement**:
- They need to invite a total of 3 men and 3 women.
- Out of these, 3 must be Ajay's relatives and 3 must be Kajol's relatives.

3. **Case Analysis**:
We will analyze the different cases based on the distribution of men and women from each side.

**Case 1**: 3 women from Ajay's side and 3 men from Kajol's side.
- Women from Ajay's side: Choose 3 from 4 women.
\[ \text{Ways} = \binom{4}{3} = 4 \]
- Men from Kajol's side: Choose 3 from 4 men.
\[ \text{Ways} = \binom{4}{3} = 4 \]
- Total for Case 1:
\[ 4 \times 4 = 16 \]

**Case 2**: 2 women from Ajay's side and 1 man from Ajay's side, and 3 women from Kajol's side.
- Women from Ajay's side: Choose 2 from 4 women.
\[ \text{Ways} = \binom{4}{2} = 6 \]
- Men from Ajay's side: Choose 1 from 3 men.
\[ \text{Ways} = \binom{3}{1} = 3 \]
- Women from Kajol's side: Choose 3 from 3 women.
\[ \text{Ways} = \binom{3}{3} = 1 \]
- Total for Case 2:
\[ 6 \times 3 \times 1 = 18 \]

**Case 3**: 1 woman from Ajay's side and 2 men from Ajay's side, and 2 women from Kajol's side and 1 man from Kajol's side.
- Women from Ajay's side: Choose 1 from 4 women.
\[ \text{Ways} = \binom{4}{1} = 4 \]
- Men from Ajay's side: Choose 2 from 3 men.
\[ \text{Ways} = \binom{3}{2} = 3 \]
- Women from Kajol's side: Choose 2 from 3 women.
\[ \text{Ways} = \binom{3}{2} = 3 \]
- Men from Kajol's side: Choose 1 from 4 men.
\[ \text{Ways} = \binom{4}{1} = 4 \]
- Total for Case 3:
\[ 4 \times 3 \times 3 \times 4 = 144 \]

**Case 4**: 0 women from Ajay's side and 3 men from Ajay's side, and 3 women from Kajol's side and 0 men from Kajol's side.
- Men from Ajay's side: Choose 3 from 3 men.
\[ \text{Ways} = \binom{3}{3} = 1 \]
- Women from Kajol's side: Choose 3 from 3 women.
\[ \text{Ways} = \binom{3}{3} = 1 \]
- Total for Case 4:
\[ 1 \times 1 = 1 \]

4. **Total Number of Ways**:
Now, we sum the total ways from all cases:
\[ 16 + 18 + 144 + 1 = 179 \]

### Final Answer:
The total number of ways they can invite 3 men and 3 women is **179**.

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