There are n persons `(n ge 3)`, among whom are A and B, who are made to stand in a row in random order. Probability that there is exactly one person between A and B is

Options

A. `(n-2)/(n(n-1))`
B. `(2(n-2))/(n(n-1))`
C. `(.^(n-2)C_(1))/(.^(n)C_(2))`
D. none of these

Answer

To find the probability that there is exactly one person between A and B when n persons are arranged in a row, we can follow these steps:

### Step 1: Total Arrangements
The total number of arrangements of n persons is given by \( n! \).

### Step 2: Favorable Arrangements
To find the number of favorable arrangements where exactly one person is between A and B, we can consider the following:

1. **Choose the person to stand between A and B**: Since A and B are fixed, we can choose any one of the remaining \( n - 2 \) persons to stand between them. This gives us \( n - 2 \) choices.

2. **Arranging A, the chosen person, and B**: The arrangement of A, the chosen person, and B can occur in two ways: either A is first and B is last (A, chosen person, B) or B is first and A is last (B, chosen person, A). Therefore, for each choice of the person in between, there are 2 arrangements.

3. **Arranging the remaining persons**: After placing A, the chosen person, and B, we have \( n - 3 \) persons left to arrange. The number of ways to arrange these \( n - 3 \) persons is \( (n - 3)! \).

Putting this all together, the number of favorable arrangements is:
\[ \text{Favorable arrangements} = (n - 2) \times 2 \times (n - 3)! \]

### Step 3: Probability Calculation
Now, we can calculate the probability \( P \) that there is exactly one person between A and B:
\[ P = \frac{\text{Favorable arrangements}}{\text{Total arrangements}} = \frac{(n - 2) \times 2 \times (n - 3)!}{n!} \]

### Step 4: Simplifying the Probability
We can simplify the expression:
\[ P = \frac{2(n - 2)(n - 3)!}{n!} \]
Since \( n! = n \times (n - 1) \times (n - 2)! \), we can rewrite it as:
\[ P = \frac{2(n - 2)(n - 3)!}{n \times (n - 1) \times (n - 2)!} \]
Now, cancel \( (n - 3)! \) from the numerator and denominator:
\[ P = \frac{2(n - 2)}{n(n - 1)} \]

### Final Result
Thus, the probability that there is exactly one person between A and B is:
\[ P = \frac{2(n - 2)}{n(n - 1)} \]

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