2m people are arranged along two sides of a long table with m chairs each side. r men wish to sit on one particular side and s on the other. In how many ways can they be seated ?`(r,s le m)`

Options

A. `"^(48)C_r`
B. 68m
C. `((2m+r)!)/(r!s!)`
D. none of these

Answer

To solve the problem of seating 2m people along two sides of a long table with m chairs on each side, where r men wish to sit on one particular side and s men on the other, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Arrangement**:
- We have 2m people and 2m chairs (m on each side of the table).
- r men want to sit on one side (let's say Side A) and s men want to sit on the other side (Side B).

2. **Choosing Seats for r Men**:
- We need to choose r chairs from the m available on Side A for the r men.
- The number of ways to choose r chairs from m is given by the combination formula:
\[ \binom{m}{r} \]

3. **Arranging r Men**:
- Once the r chairs are chosen, the r men can be arranged in those chairs.
- The number of ways to arrange r men is given by r factorial (r!):
\[ r! \]

4. **Choosing Seats for s Men**:
- Similarly, we need to choose s chairs from the m available on Side B for the s men.
- The number of ways to choose s chairs from m is:
\[ \binom{m}{s} \]

5. **Arranging s Men**:
- Once the s chairs are chosen, the s men can be arranged in those chairs.
- The number of ways to arrange s men is given by s factorial (s!):
\[ s! \]

6. **Seating Remaining People**:
- After seating r men on Side A and s men on Side B, we have:
\[ 2m - r - s \]
people left to be seated.
- These remaining people can sit in any of the remaining chairs (m - r on Side A and m - s on Side B).
- The total number of ways to arrange these remaining people is:
\[ (2m - r - s)! \]

7. **Combining All Parts**:
- The total number of ways to arrange all the people is the product of all the combinations and arrangements:
\[ \text{Total Ways} = \binom{m}{r} \times r! \times \binom{m}{s} \times s! \times (2m - r - s)! \]

### Final Formula:
Thus, the total number of ways to seat the people is:
\[ \text{Total Ways} = \binom{m}{r} \times r! \times \binom{m}{s} \times s! \times (2m - r - s)! \]