Fifteen persons, among whom are A and B, sit down at random on a round table. The probability that there are 4 persons between A and B, is

Options

A. `(9 !)/(14 !)`
B. `(10 !)/(14 !)`
C. `(9 !)/(15 !)`
D. none of these

Explanation

To solve the problem of finding the probability that there are 4 persons between A and B when 15 persons sit down at random at a round table, we can follow these steps:

### Step 1: Calculate the total arrangements of 15 persons around a round table.
In circular permutations, the number of ways to arrange \( n \) persons is given by \( (n-1)! \). Therefore, for 15 persons, the total arrangements are:
\[ 14! \]

**Hint:** Remember that in a circular arrangement, we fix one person to avoid counting rotations as different arrangements.

### Step 2: Calculate the favorable arrangements where there are 4 persons between A and B.
To find the number of arrangements where A and B have exactly 4 persons between them, we can visualize the arrangement as follows:

1. Fix A in one position.
2. The positions around A can be labeled as follows:
- A, P1, P2, P3, P4, B (where P1, P2, P3, and P4 are the 4 persons between A and B)
- A, B, P1, P2, P3, P4 (the reverse arrangement)

This means we can have two configurations: A on one side and B on the other, or vice versa.

### Step 3: Choose 4 persons from the remaining 13 persons.
Since A and B are fixed, we have 13 persons left. We need to choose 4 persons from these 13 to sit between A and B. The number of ways to choose 4 persons from 13 is given by:
\[ \binom{13}{4} \]

**Hint:** Use the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \) to calculate the number of ways to choose r persons from n persons.

### Step 4: Arrange the chosen 4 persons.
The 4 persons chosen can be arranged in \( 4! \) ways.

### Step 5: Arrange the remaining 9 persons.
After placing A, B, and the 4 chosen persons, we have 9 persons left. These 9 persons can be arranged in \( 9! \) ways.

### Step 6: Consider the arrangements of A and B.
Since A and B can be arranged in 2 ways (A can be to the left of B or B can be to the left of A), we multiply the total arrangements by 2.

### Step 7: Calculate the total favorable arrangements.
Putting it all together, the total number of favorable arrangements is:
\[ \text{Favorable cases} = 2 \times \binom{13}{4} \times 4! \times 9! \]

### Step 8: Calculate the probability.
The probability \( P \) that there are 4 persons between A and B is given by the ratio of favorable cases to total arrangements:
\[ P = \frac{\text{Favorable cases}}{\text{Total arrangements}} = \frac{2 \times \binom{13}{4} \times 4! \times 9!}{14!} \]

### Step 9: Simplify the expression.
Using the fact that \( 14! = 14 \times 13! \), we can simplify:
\[ P = \frac{2 \times \binom{13}{4} \times 4! \times 9!}{14 \times 13!} \]

Since \( \binom{13}{4} = \frac{13!}{4! \times 9!} \), we can substitute this into the equation:
\[ P = \frac{2 \times \frac{13!}{4! \times 9!} \times 4! \times 9!}{14 \times 13!} = \frac{2}{14} = \frac{1}{7} \]

### Final Answer:
Thus, the probability that there are 4 persons between A and B is:
\[ \frac{1}{7} \]