A and B stand in a ring with 10 other persons . If the arrangement of the twelve persons is at random , find the chance that there are exactly three persons between A and B.

Answer

To solve the problem of finding the probability that there are exactly three persons between A and B in a circular arrangement of 12 persons, we can follow these steps:

### Step 1: Understand the arrangement
In a circular arrangement of 12 persons (A, B, and 10 others), we need to find the number of arrangements where exactly three persons are between A and B.

### Step 2: Fix A's position
Since the arrangement is circular, we can fix A's position to simplify our calculations. This means we can consider A to be at a specific point in the circle.

### Step 3: Identify positions for B
If A is fixed, B can be positioned in one of two ways to have exactly three persons between them:
- B can be placed 4 positions clockwise from A.
- B can be placed 4 positions counterclockwise from A.

### Step 4: Choose 3 persons from the remaining 10
Now, we need to select 3 persons from the remaining 10 persons to be placed between A and B. The number of ways to choose 3 persons from 10 is given by the combination formula:
\[ \binom{10}{3} \]

### Step 5: Arrange the 3 chosen persons
The 3 chosen persons can be arranged in the 3 positions between A and B in \(3!\) (factorial of 3) ways.

### Step 6: Arrange the remaining persons
After placing A, B, and the 3 persons between them, we have 7 persons left (10 total - 3 chosen = 7 remaining). These 7 persons can be arranged in the remaining 7 positions in \(7!\) ways.

### Step 7: Calculate the total arrangements
Now, we can calculate the total number of favorable arrangements:
\[ \text{Favorable arrangements} = 2 \times \binom{10}{3} \times 3! \times 7! \]
The factor of 2 accounts for the two possible positions of B relative to A.

### Step 8: Calculate total arrangements in a circle
The total number of arrangements of 12 persons in a circle is given by:
\[ (12 - 1)! = 11! \]

### Step 9: Calculate the probability
The probability \(P\) that there are exactly 3 persons between A and B is given by:
\[ P = \frac{\text{Favorable arrangements}}{\text{Total arrangements}} = \frac{2 \times \binom{10}{3} \times 3! \times 7!}{11!} \]

### Step 10: Simplify the expression
Now we can substitute the values:
\[ \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]
Thus,
\[ Favorable arrangements = 2 \times 120 \times 6 \times 5040 \]
Calculating this gives:
\[ Favorable arrangements = 2 \times 120 \times 6 \times 5040 = 7257600 \]
And since \(11! = 39916800\), we can find the probability:
\[ P = \frac{7257600}{39916800} = \frac{2}{11} \]

### Final Answer
Thus, the probability that there are exactly three persons between A and B is:
\[ \boxed{\frac{2}{11}} \]