A committee of 5 is to be formed from a group of 10 people consisting 4 single men 4 single women and a married couple. The committee is to consist of a chairman , who must be a single man 2, other men and 2 women,
How many of these would include the married couples?

Answer

To solve the problem, we need to form a committee of 5 people from a group of 10, which includes 4 single men, 4 single women, and 1 married couple. The committee must consist of a chairman (who must be a single man), 2 other men, and 2 women. We also need to find how many of these committees include the married couple.

### Step-by-Step Solution:

1. **Choose the Chairman:**
The chairman must be one of the 4 single men.
- The number of ways to choose the chairman = \( \binom{4}{1} = 4 \).

2. **Choose the Other Men:**
After selecting the chairman, we need to choose 2 other men from the remaining men. Since one single man has already been chosen as chairman, there are now 3 single men left.
- The number of ways to choose 2 other men from these 3 = \( \binom{3}{2} = 3 \).

3. **Choose the Women:**
We need to choose 2 women from the 4 single women available.
- The number of ways to choose 2 women = \( \binom{4}{2} = 6 \).

4. **Calculate Total Committees:**
Now, we can calculate the total number of ways to form the committee:
\[ \text{Total Committees} = \text{Ways to choose Chairman} \times \text{Ways to choose Other Men} \times \text{Ways to choose Women} \]
\[ \text{Total Committees} = 4 \times 3 \times 6 = 72 \]

5. **Including the Married Couple:**
To find the number of committees that include the married couple, we need to account for the fact that if the married couple is included, they take up 2 spots in the committee.

- **Choosing the Chairman:** The chairman can still be one of the 4 single men, so the number of ways to choose the chairman remains \( \binom{4}{1} = 4 \).

- **Choosing the Other Men:** Since the married couple takes up 2 spots, we need to choose 1 more man from the remaining 3 single men (after the chairman is chosen).
- The number of ways to choose 1 more man = \( \binom{3}{1} = 3 \).

- **Choosing the Women:** Since the married couple takes up 2 spots, we still need to choose 2 women from the 4 single women.
- The number of ways to choose 2 women = \( \binom{4}{1} = 4 \).

6. **Calculate Total Committees Including Married Couple:**
Now we can calculate the total number of committees that include the married couple:
\[ \text{Total Committees with Married Couple} = \text{Ways to choose Chairman} \times \text{Ways to choose Other Man} \times \text{Ways to choose Women} \]
\[ \text{Total Committees with Married Couple} = 4 \times 3 \times 4 = 48 \]

### Final Answer:
The total number of committees that include the married couple is **48**.