7 relative of a man comprises 4 ladies and 3 gentleman, his wife has also 7 relatives. 3 of them are ladies and 4 gentlemen. In how ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of man's relative and 3 of the wife's relatives.

Explanation

To solve the problem of inviting a dinner party of 3 ladies and 3 gentlemen from the relatives of a man and his wife, we can break down the solution into clear steps.

### Step-by-Step Solution:

1. **Identify the relatives**:
- The man has 4 ladies and 3 gentlemen.
- The wife has 3 ladies and 4 gentlemen.

2. **Determine the combinations**:
- We need to invite a total of 6 guests: 3 ladies and 3 gentlemen.
- The guests must consist of 3 relatives from the man and 3 relatives from the wife.

3. **Consider different cases**:
- We can have different combinations of ladies and gentlemen from both sides. The cases can be categorized as follows:
- Case 1: 0 ladies from the man's side, 3 gentlemen from the man's side.
- Case 2: 1 lady from the man's side, 2 gentlemen from the man's side.
- Case 3: 2 ladies from the man's side, 1 gentleman from the man's side.
- Case 4: 3 ladies from the man's side, 0 gentlemen from the man's side.

4. **Calculate the number of ways for each case**:
- **Case 1**:
- 0 ladies from the man's side (choose from 4): \( \binom{4}{0} = 1 \)
- 3 gentlemen from the man's side (choose from 3): \( \binom{3}{3} = 1 \)
- 3 ladies from the wife's side (choose from 3): \( \binom{3}{3} = 1 \)
- 0 gentlemen from the wife's side (choose from 4): \( \binom{4}{0} = 1 \)
- Total for Case 1: \( 1 \times 1 \times 1 \times 1 = 1 \)

- **Case 2**:
- 1 lady from the man's side (choose from 4): \( \binom{4}{1} = 4 \)
- 2 gentlemen from the man's side (choose from 3): \( \binom{3}{2} = 3 \)
- 2 ladies from the wife's side (choose from 3): \( \binom{3}{2} = 3 \)
- 1 gentleman from the wife's side (choose from 4): \( \binom{4}{1} = 4 \)
- Total for Case 2: \( 4 \times 3 \times 3 \times 4 = 144 \)

- **Case 3**:
- 2 ladies from the man's side (choose from 4): \( \binom{4}{2} = 6 \)
- 1 gentleman from the man's side (choose from 3): \( \binom{3}{1} = 3 \)
- 1 lady from the wife's side (choose from 3): \( \binom{3}{1} = 3 \)
- 2 gentlemen from the wife's side (choose from 4): \( \binom{4}{2} = 6 \)
- Total for Case 3: \( 6 \times 3 \times 3 \times 6 = 324 \)

- **Case 4**:
- 3 ladies from the man's side (choose from 4): \( \binom{4}{3} = 4 \)
- 0 gentlemen from the man's side (choose from 3): \( \binom{3}{0} = 1 \)
- 0 ladies from the wife's side (choose from 3): \( \binom{3}{0} = 1 \)
- 3 gentlemen from the wife's side (choose from 4): \( \binom{4}{3} = 4 \)
- Total for Case 4: \( 4 \times 1 \times 1 \times 4 = 16 \)

5. **Sum the total ways**:
- Total ways = Case 1 + Case 2 + Case 3 + Case 4
- Total ways = \( 1 + 144 + 324 + 16 = 485 \)

### Final Answer:
The total number of ways to invite the dinner party is **485**.