The number of ways in which a mixed doubles game in tennis can be arranged form 5 married couples, if no husband and wife play in the same game, is

Options

A. A. 100
B. B. 60
C. C. 80
D. D. 50

Answer

To solve the problem of arranging a mixed doubles game in tennis from 5 married couples, where no husband and wife play in the same game, we can follow these steps:

### Step 1: Select the Husbands
We need to select 2 husbands from the 5 available husbands. The number of ways to choose 2 husbands from 5 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

\[ \text{Number of ways to select 2 husbands} = \binom{5}{2} \]

### Step 2: Select the Wives
Since no husband can play with his wife, after selecting 2 husbands, we cannot select their wives. This leaves us with 3 wives to choose from. We need to select 2 wives from these 3 remaining wives.

\[ \text{Number of ways to select 2 wives} = \binom{3}{2} \]

### Step 3: Calculate the Combinations
Now we will calculate the combinations we found in the previous steps.

1. Calculate \( \binom{5}{2} \):
\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

2. Calculate \( \binom{3}{2} \):
\[ \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2}{2 \times 1} = 3 \]

### Step 4: Multiply the Combinations
Now, we multiply the number of ways to select the husbands by the number of ways to select the wives:

\[ \text{Total ways} = \binom{5}{2} \times \binom{3}{2} = 10 \times 3 = 30 \]

### Step 5: Arrange the Selected Players
In a mixed doubles game, the arrangement of the selected husbands and wives matters. Since we have selected 2 husbands and 2 wives, we can arrange them in \( 2! \) ways (for the husbands) and \( 2! \) ways (for the wives).

\[ \text{Arrangements} = 2! \times 2! = 2 \times 2 = 4 \]

### Step 6: Final Calculation
Now we multiply the total ways to select the players by the arrangements:

\[ \text{Total arrangements} = 30 \times 4 = 120 \]

Thus, the total number of ways to arrange a mixed doubles game from 5 married couples, ensuring no husband and wife play together, is **120**.