In how many ways can a mixed doubles game in tennis be arranged from 5 married couples, if no husband and wife play in the same game and two males are always opponents to two females?

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 and two males are always opponents to two females, we can follow these steps:

### Step 1: Choose 2 males from the 5 males.
We can select 2 males from the 5 males using the combination formula \( nCr \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

\[ \text{Number of ways to choose 2 males} = \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 2: Identify the females that cannot play.
Once we have selected 2 males, their respective wives cannot participate in the game. This leaves us with 3 females to choose from.

### Step 3: Choose 2 females from the remaining 3 females.
Now we need to select 2 females from the 3 remaining females.

\[ \text{Number of ways to choose 2 females} = \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2}{2 \times 1} = 3 \]

### Step 4: Calculate the total arrangements.
Now, we can find the total number of ways to arrange the mixed doubles game by multiplying the number of ways to choose the males and the females.

\[ \text{Total arrangements} = \text{Number of ways to choose 2 males} \times \text{Number of ways to choose 2 females} = 10 \times 3 = 30 \]

### Final Answer:
Thus, the total number of ways to arrange the mixed doubles game is **30**.

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