A man has 5 male and 4 female relatives. His wife has 4 male and 5 female relatives. The number of ways in which they can invite 5 male and 5 female relatives so that 5 of them are man's relatives and remaining 5 are his wife's relatives

Options

A. A 5426
B. B 5226
C. C 5526
D. D 5626

Answer

To solve the problem of how many ways a man and his wife can invite 5 male and 5 female relatives, with the condition that 5 of them are the man's relatives and the remaining 5 are his wife's relatives, we can break it down into steps.

### Step-by-Step Solution:

1. **Identify the Relatives**:
- The man has 5 male and 4 female relatives.
- The wife has 4 male and 5 female relatives.

2. **Determine the Combinations for Male Relatives**:
- We need to select a total of 5 male relatives. The combinations can be:
- 5 males from the man and 0 from the wife.
- 4 males from the man and 1 from the wife.
- 3 males from the man and 2 from the wife.
- 2 males from the man and 3 from the wife.
- 1 male from the man and 4 from the wife.
- 0 males from the man and 5 from the wife.

3. **Calculate Each Case**:
- **Case 1**: 5 males from the man (5C5) and 0 from the wife (4C0).
\[ \text{Ways} = 1 \times 1 = 1 \]
- **Case 2**: 4 males from the man (5C4) and 1 from the wife (4C1).
\[ \text{Ways} = 5 \times 4 = 20 \]
- **Case 3**: 3 males from the man (5C3) and 2 from the wife (4C2).
\[ \text{Ways} = 10 \times 6 = 60 \]
- **Case 4**: 2 males from the man (5C2) and 3 from the wife (4C3).
\[ \text{Ways} = 10 \times 4 = 40 \]
- **Case 5**: 1 male from the man (5C1) and 4 from the wife (4C4).
\[ \text{Ways} = 5 \times 1 = 5 \]
- **Case 6**: 0 males from the man (5C0) and 5 from the wife (4C5).
\[ \text{Ways} = 1 \times 0 = 0 \]

4. **Sum the Ways for Male Relatives**:
\[ \text{Total Ways (Male)} = 1 + 20 + 60 + 40 + 5 + 0 = 126 \]

5. **Determine the Combinations for Female Relatives**:
- Similar to the males, we need to select a total of 5 female relatives. The combinations can be:
- 5 females from the man and 0 from the wife.
- 4 females from the man and 1 from the wife.
- 3 females from the man and 2 from the wife.
- 2 females from the man and 3 from the wife.
- 1 female from the man and 4 from the wife.
- 0 females from the man and 5 from the wife.

6. **Calculate Each Case for Female Relatives**:
- **Case 1**: 5 females from the man (4C5) and 0 from the wife (5C0).
\[ \text{Ways} = 0 \times 1 = 0 \]
- **Case 2**: 4 females from the man (4C4) and 1 from the wife (5C1).
\[ \text{Ways} = 1 \times 5 = 5 \]
- **Case 3**: 3 females from the man (4C3) and 2 from the wife (5C2).
\[ \text{Ways} = 4 \times 10 = 40 \]
- **Case 4**: 2 females from the man (4C2) and 3 from the wife (5C3).
\[ \text{Ways} = 6 \times 10 = 60 \]
- **Case 5**: 1 female from the man (4C1) and 4 from the wife (5C4).
\[ \text{Ways} = 4 \times 5 = 20 \]
- **Case 6**: 0 females from the man (4C0) and 5 from the wife (5C5).
\[ \text{Ways} = 1 \times 1 = 1 \]

7. **Sum the Ways for Female Relatives**:
\[ \text{Total Ways (Female)} = 0 + 5 + 40 + 60 + 20 + 1 = 126 \]

8. **Calculate Total Combinations**:
\[ \text{Total Combinations} = \text{Total Ways (Male)} \times \text{Total Ways (Female)} = 126 \times 126 = 15876 \]

### Final Answer:
The total number of ways they can invite 5 male and 5 female relatives is **15876**.