The number of ways in which a mixed doubles game can be arranged from 9 married couples if no husband and wife play in the same game is:

Options

A. 756
B. 3024
C. 1512
D. none

Answer

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

### Step 1: Understand the Requirements
In a mixed doubles game, each team consists of one male and one female. Since we have 9 married couples, we have 9 males and 9 females. The condition is that no husband and wife can play together.

### Step 2: Select the Males
We need to select 2 males from the 9 available males. The number of ways to choose 2 males from 9 is given by the combination formula:
\[ \text{Number of ways to choose males} = \binom{9}{2} \]

### Step 3: Select the Females
Since no husband can play with his wife, after selecting 2 males, we cannot select their wives. This leaves us with 7 females to choose from. We need to select 2 females from these 7:
\[ \text{Number of ways to choose females} = \binom{7}{2} \]

### Step 4: Calculate the Combinations
Now, we will calculate the combinations:
- For males:
\[ \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \]
- For females:
\[ \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \]

### Step 5: Consider the Arrangement of Teams
In a mixed doubles game, the two selected males can pair with the two selected females in different ways. Each male can be paired with each female, so we have 2 arrangements (since the order matters):
\[ \text{Number of arrangements} = 2 \]

### Step 6: Calculate the Total Number of Ways
Now, we can find the total number of ways to arrange the mixed doubles game:
\[ \text{Total ways} = \binom{9}{2} \times \binom{7}{2} \times 2 = 36 \times 21 \times 2 \]

### Step 7: Perform the Final Calculation
Calculating the total:
\[ 36 \times 21 = 756 \]
\[ 756 \times 2 = 1512 \]

### Conclusion
Thus, the total number of ways to arrange a mixed doubles game from 9 married couples, ensuring no husband and wife play together, is:
\[ \boxed{1512} \]