How many different group can be selected for playing tennis out of 4 ladies and 3 gentalemen, there being one lady and one gentleman on each side?

Answer

To solve the problem of how many different groups can be selected for playing tennis out of 4 ladies and 3 gentlemen, where there is one lady and one gentleman on each side, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the number of choices for ladies**:
We have 4 ladies, and we need to select 2 ladies (one for each team). The number of ways to choose 2 ladies from 4 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. Thus, we calculate:
\[ \text{Number of ways to choose ladies} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

2. **Identify the number of choices for gentlemen**:
Similarly, we have 3 gentlemen and need to select 2 gentlemen (one for each team). The number of ways to choose 2 gentlemen from 3 is:
\[ \text{Number of ways to choose gentlemen} = \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2}{2 \times 1} = 3 \]

3. **Calculate the total combinations for teams**:
Now, we need to find the total number of ways to select one lady and one gentleman for each team. Since the selections are independent, we multiply the number of ways to choose the ladies by the number of ways to choose the gentlemen:
\[ \text{Total ways} = \text{Ways to choose ladies} \times \text{Ways to choose gentlemen} = 6 \times 3 = 18 \]

4. **Consider interchanging players**:
Since the players can interchange their teams, we need to account for this. Each selection of 1 lady and 1 gentleman can be arranged in two ways (Team A and Team B can switch). Therefore, we multiply the total combinations by 2:
\[ \text{Total arrangements} = 2 \times 18 = 36 \]

### Final Answer:
Thus, the total number of different groups that can be selected for playing tennis is **36**.

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