Ten players are to be seated in a row for photographs , so that the two particular players sit in the 2 middle seats. The number of arrangements is

Options

A. `9!`
B. `(9!)(2!)`
C. `2.(8!)`
D. none of these

Answer

To solve the problem of arranging 10 players in a row such that two particular players sit in the two middle seats, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Middle Seats**:
In a row of 10 seats, the middle seats are the 5th and 6th seats. We need to place the two particular players, let's call them Player A and Player B, in these seats.

2. **Arrange Players A and B**:
There are 2 ways to arrange Players A and B in the middle seats. Player A can sit in the 5th seat and Player B in the 6th seat, or Player A can sit in the 6th seat and Player B in the 5th seat. Thus, the arrangements for Players A and B are:
\[ 2! = 2 \text{ ways} \]

3. **Arrange the Remaining Players**:
After placing Players A and B, we have 8 remaining players who need to be seated in the remaining 8 seats (1st, 2nd, 3rd, 4th, 7th, 8th, 9th, and 10th). The number of ways to arrange these 8 players is given by the factorial of the number of players:
\[ 8! = 40320 \text{ ways} \]

4. **Calculate Total Arrangements**:
To find the total number of arrangements, we multiply the number of arrangements of Players A and B by the number of arrangements of the remaining players:
\[ \text{Total Arrangements} = 2! \times 8! = 2 \times 40320 = 80640 \]

### Final Answer:
The total number of arrangements of the 10 players, with the condition that Players A and B sit in the middle seats, is **80,640**.

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