A party of n ladies sit at a round table. Find odds against two specified ladies sitting next to each other

Options

A. `2 : n-3`
B. `n-3 : 2`
C. `n-2 : 2`
D. `2 : n-2`

Answer

To solve the problem of finding the odds against two specified ladies sitting next to each other at a round table with n ladies, we can follow these steps:

### Step 1: Calculate the Total Arrangements
The total number of ways to arrange n ladies at a round table is given by the formula:
\[ (n - 1)! \]
This is because one lady can be fixed to break the circular symmetry, and the remaining (n - 1) ladies can be arranged around her.

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

### Step 2: Treat Two Specified Ladies as a Single Unit
When considering the two specified ladies (let's call them L1 and L2) sitting next to each other, we can treat them as a single unit or block. This means we now have (n - 1) units to arrange: the block of L1 and L2, and the remaining (n - 2) ladies.

**Hint:** When treating two people as a single unit, always account for the remaining individuals in the arrangement.

### Step 3: Calculate Arrangements with L1 and L2 Together
The number of ways to arrange these (n - 1) units in a round table is:
\[ (n - 2)! \]
Additionally, within their block, L1 and L2 can switch places, which gives us an extra factor of \(2!\) (or 2) for their arrangements.

Thus, the total arrangements with L1 and L2 sitting together is:
\[ (n - 2)! \times 2! \]

**Hint:** Always consider internal arrangements when treating a group as a single unit.

### Step 4: Calculate the Probability of L1 and L2 Sitting Together
The probability \(P(T)\) that L1 and L2 are sitting together is given by the ratio of the favorable outcomes to the total outcomes:
\[ P(T) = \frac{(n - 2)! \times 2!}{(n - 1)!} \]
This simplifies to:
\[ P(T) = \frac{2}{n - 1} \]

**Hint:** When calculating probabilities, ensure you simplify the fractions correctly.

### Step 5: Calculate the Probability of L1 and L2 Not Sitting Together
The probability \(P(N)\) that L1 and L2 are not sitting together can be found by subtracting the probability of them sitting together from 1:
\[ P(N) = 1 - P(T) = 1 - \frac{2}{n - 1} = \frac{n - 3}{n - 1} \]

**Hint:** The complement rule is useful for finding probabilities of "not" events.

### Step 6: Calculate the Odds Against L1 and L2 Sitting Together
The odds against an event is given by the ratio of the probability of the event not occurring to the probability of the event occurring:
\[ \text{Odds against} = \frac{P(N)}{P(T)} = \frac{\frac{n - 3}{n - 1}}{\frac{2}{n - 1}} = \frac{n - 3}{2} \]

**Hint:** When calculating odds, remember to divide the probabilities correctly.

### Final Result
Thus, the odds against the two specified ladies sitting next to each other is:
\[ \text{Odds against} = \frac{n - 3}{2} \]