In how many ways can a mixed double game can be arranged from amongst 9 married couples if no husband and wife play in the same game?

Options

A. a. 2840
B. b 3024
C. c. 4800
D. d. none of these

Answer

To solve the problem of arranging a mixed doubles game from 9 married couples with the condition that no husband and wife can play in the same game, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 9 married couples, which means there are 9 men and 9 women. We need to select 2 men and 2 women for a mixed doubles game, ensuring that the selected men are not paired with their wives.

2. **Selecting Men**: First, we need to choose 2 men from the 9 available men. The number of ways to choose 2 men from 9 can be calculated using the combination formula:
\[ \text{Number of ways to choose 2 men} = \binom{9}{2} \]
Calculating this gives:
\[ \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \]

3. **Selecting Women**: After selecting 2 men, we cannot select their wives. Therefore, we have 7 women left to choose from (since 2 women are excluded). The number of ways to choose 2 women from these 7 is:
\[ \text{Number of ways to choose 2 women} = \binom{7}{2} \]
Calculating this gives:
\[ \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \]

4. **Pairing the Selected Men and Women**: Once we have selected 2 men and 2 women, we can pair them in 2 different ways. For example, if we selected men A and B and women C and D, the pairings can be (A with C and B with D) or (A with D and B with C). Thus, there are 2 ways to pair them.

5. **Calculating the Total Arrangements**: Now we can calculate the total number of ways to arrange the mixed doubles game by multiplying the number of ways to choose the men, the number of ways to choose the women, and the number of ways to pair them:
\[ \text{Total arrangements} = \binom{9}{2} \times \binom{7}{2} \times 2 \]
Substituting the values we calculated:
\[ \text{Total arrangements} = 36 \times 21 \times 2 = 1512 \]

### Final Answer:
The total number of ways to arrange a mixed doubles game from 9 married couples, ensuring no husband and wife play together, is **1512**.