The number of ways in which two teams A and B of 11 players each can be made up from 22 players so that two particular players are on the opposite sides is:

Options

A. 369512
B. 184755
C. 184756
D. 369514

Answer

To solve the problem of forming two teams A and B of 11 players each from a pool of 22 players, with the condition that two specific players (let's call them P1 and P2) must be on opposite teams, we can follow these steps:

### Step-by-Step Solution:

1. **Assign Players to Teams**:
- We start by placing player P1 in Team A and player P2 in Team B. This ensures that they are on opposite teams, as required by the problem.

2. **Calculate Remaining Players**:
- After placing P1 and P2, we have 22 - 2 = 20 players left to choose from.

3. **Select Additional Players for Each Team**:
- Since we already have 1 player in each team, we need to select 10 more players for Team A and 10 more players for Team B.
- Therefore, we need to choose 10 players for Team A from the remaining 20 players. The number of ways to do this is given by the combination formula:
\[ \binom{20}{10} \]

4. **Determine Remaining Players for Team B**:
- After selecting 10 players for Team A, the remaining players will automatically go to Team B. Since we have chosen 10 players for Team A from 20, the remaining 10 players will go to Team B.

5. **Account for Player Interchange**:
- Since we could have also placed P2 in Team A and P1 in Team B, we need to account for this interchange. There are 2 ways to arrange P1 and P2 in their respective teams (P1 in A and P2 in B, or P1 in B and P2 in A). Thus, we multiply our result by 2.

6. **Final Calculation**:
- The total number of ways to form the teams is:
\[ 2 \times \binom{20}{10} \]

7. **Calculate the Combination**:
- Now we calculate \(\binom{20}{10}\):
\[ \binom{20}{10} = \frac{20!}{10! \times 10!} \]
- Therefore, the total number of ways is:
\[ 2 \times \frac{20!}{10! \times 10!} \]

### Final Answer:
- The total number of ways to form the two teams such that P1 and P2 are on opposite teams is:
\[ 2 \times \binom{20}{10} = 2 \times 184756 = 369512 \]