A party of n men is to be seated round a circular table . Find the odds against the event two particular men sit together .

Answer

To solve the problem of finding the odds against two particular men sitting together at a circular table with \( n \) men, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Circular Permutations**:
When arranging \( n \) people around a circular table, the total number of arrangements (permutations) is given by \( (n-1)! \).

**Hint**: Remember that in circular permutations, one position is fixed to avoid counting rotations as different arrangements.

2. **Defining the Event**:
Let’s denote the two particular men as \( A \) and \( B \). We want to find the arrangements where \( A \) and \( B \) sit together.

3. **Treating \( A \) and \( B \) as a Single Unit**:
If \( A \) and \( B \) are sitting together, we can treat them as a single unit or block. This means we now have \( n-1 \) units to arrange: the block \( (AB) \) and the remaining \( n-2 \) men.

**Hint**: Think of \( A \) and \( B \) as a single entity or block to simplify the arrangement.

4. **Calculating Arrangements with \( A \) and \( B \) Together**:
The number of ways to arrange these \( n-1 \) units in a circular manner is \( (n-2)! \). Additionally, within the block \( (AB) \), \( A \) and \( B \) can switch places, giving us an extra factor of \( 2 \).

Therefore, the total arrangements where \( A \) and \( B \) sit together is:
\[ \text{Favorable arrangements} = 2 \times (n-2)! \]

5. **Calculating the Probability of \( A \) and \( B \) Sitting Together**:
The probability \( P(A) \) that \( A \) and \( B \) sit together is given by the ratio of favorable arrangements to total arrangements:
\[ P(A) = \frac{2 \times (n-2)!}{(n-1)!} = \frac{2}{n-1} \]

**Hint**: Remember to simplify the factorial expressions carefully.

6. **Calculating the Probability of \( A \) and \( B \) Not Sitting Together**:
The probability \( P(A') \) that \( A \) and \( B \) do not sit together is:
\[ P(A') = 1 - P(A) = 1 - \frac{2}{n-1} = \frac{n-3}{n-1} \]

7. **Finding the Odds Against \( A \) and \( B \) Sitting Together**:
The odds against \( A \) and \( B \) sitting together can be expressed as the ratio of the probability of them not sitting together to the probability of them sitting together:
\[ \text{Odds against} = \frac{P(A')}{P(A)} = \frac{\frac{n-3}{n-1}}{\frac{2}{n-1}} = \frac{n-3}{2} \]

### Final Result:
Thus, the odds against the event that two particular men sit together at a circular table with \( n \) men is:
\[ \text{Odds against} = \frac{n-3}{2} \]