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 break down the solution into clear steps.

### Step-by-Step Solution:

1. **Identify the Players**: We have a total of 22 players. Let's denote the two particular players as P1 and P2.

2. **Assign Teams for P1 and P2**: Since P1 and P2 must be on opposite teams, we can assign P1 to Team A and P2 to Team B. This ensures that they are on different teams.

3. **Select Remaining Players for Team A**: After assigning P1 to Team A, we have 21 players left (22 total - 1 for P1). Since Team A already has P1, we need to choose 10 more players from the remaining 20 players (excluding P2).

The number of ways to choose 10 players from 20 is given by the combination formula:
\[ \binom{20}{10} \]

4. **Select Remaining Players for Team B**: After selecting 10 players for Team A, Team B will automatically consist of the remaining players. Since P2 is already in Team B, we will have 11 players in Team B (P2 + the 10 players not chosen for Team A).

5. **Calculate the Total Combinations**: The total number of ways to form the teams, considering the assignment of P1 and P2 and the selection of the remaining players, is:
\[ \text{Total Ways} = \binom{20}{10} \]

6. **Account for the Swapping of P1 and P2**: Since P1 and P2 could be swapped (i.e., P1 could be in Team B and P2 in Team A), we multiply the total combinations by 2:
\[ \text{Total Ways} = 2 \times \binom{20}{10} \]

7. **Calculate the Value of \(\binom{20}{10}\)**:
\[ \binom{20}{10} = \frac{20!}{10! \times 10!} = 184756 \]

8. **Final Calculation**:
\[ \text{Total Ways} = 2 \times 184756 = 369512 \]

### Final Answer:
The total number of ways to form the two teams such that P1 and P2 are on opposite sides is **369512**.