Six married couple are sitting in a room. Number of ways in which 4 people can be selected so that there is exactly one married couple among the four is:

Options

A. 276
B. 600
C. 840
D. 240

Answer

To solve the problem of selecting 4 people from 6 married couples such that exactly one married couple is included, we can follow these steps:

### Step 1: Choose the Married Couple
We first need to select one married couple from the 6 available couples. The number of ways to choose 1 couple from 6 is given by the combination formula:

\[ \text{Number of ways to choose 1 couple} = \binom{6}{1} = 6 \]

**Hint:** Think of the couples as pairs, and you are simply selecting one pair from the total.

### Step 2: Select the Remaining 2 People
After selecting one couple, we need to select 2 more individuals from the remaining 10 people (5 men and 5 women). However, we need to ensure that we do not select the spouse of the man chosen from the couple.

We can break this down into three cases for selecting the remaining 2 people:

1. **Case 1:** Both selected are men (not including the husband from the selected couple).
2. **Case 2:** Both selected are women (not including the wife from the selected couple).
3. **Case 3:** One selected is a man (not including the husband from the selected couple) and one is a woman (not including the wife from the selected couple).

#### Case 1: Selecting 2 Men
We have 5 men left (excluding the selected husband). The number of ways to select 2 men from these 5 is:

\[ \text{Number of ways} = \binom{5}{2} = 10 \]

#### Case 2: Selecting 2 Women
Similarly, we have 5 women left (excluding the selected wife). The number of ways to select 2 women from these 5 is:

\[ \text{Number of ways} = \binom{5}{2} = 10 \]

#### Case 3: Selecting 1 Man and 1 Woman
We can select 1 man from the remaining 5 men and 1 woman from the remaining 4 women (since we cannot select the wife of the selected husband). The number of ways to do this is:

\[ \text{Number of ways} = \binom{5}{1} \times \binom{4}{1} = 5 \times 4 = 20 \]

### Step 3: Combine the Cases
Now we can combine the results from all three cases to find the total number of ways to select the remaining 2 people:

\[ \text{Total ways to select remaining 2 people} = 10 + 10 + 20 = 40 \]

### Step 4: Calculate the Total Combinations
Finally, we multiply the number of ways to choose the couple by the number of ways to select the remaining individuals:

\[ \text{Total selections} = \text{Ways to choose couple} \times \text{Ways to select remaining 2} = 6 \times 40 = 240 \]

Thus, the total number of ways to select 4 people such that exactly one married couple is included is **240**.

### Summary
The final answer is:

\[ \boxed{240} \]