12 members of a committee are to sit down at random round a table. Probability that there are 3 members between the two particular members A and B, is

Options

A. `2/11`
B. `7/11`
C. `4/11`
D. `5/11`

Answer

To solve the problem of finding the probability that there are exactly 3 members between two particular members A and B when 12 members are seated randomly around a round table, we can follow these steps:

### Step 1: Total Arrangements
When arranging \( n \) members around a round table, the total number of arrangements is given by \( (n-1)! \). For 12 members, the total arrangements are:
\[ \text{Total arrangements} = 11! \]

**Hint:** Remember that in circular arrangements, one position is fixed to avoid counting identical arrangements due to rotation.

### Step 2: Fixing Positions for A and B
We need to find the arrangements where there are exactly 3 members between A and B. We can visualize the arrangement as follows:
- If A is seated, B must be seated 4 positions away (3 members in between).

**Hint:** Think of A and B as fixed points in the arrangement to simplify counting.

### Step 3: Arranging A and B
Since A and B can switch places, we have 2 arrangements for A and B:
\[ \text{Arrangements for A and B} = 2 \]

**Hint:** Consider both configurations (A first or B first) when counting arrangements.

### Step 4: Choosing Members Between A and B
After fixing A and B, we need to choose 3 members from the remaining 10 members to sit between them. The number of ways to choose 3 members from 10 is given by:
\[ \text{Ways to choose 3 members} = \binom{10}{3} \]

**Hint:** Use the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).

### Step 5: Arranging the 3 Members
The 3 members chosen can be arranged among themselves in:
\[ \text{Arrangements of 3 members} = 3! \]

**Hint:** Remember that the order matters when arranging the members.

### Step 6: Arranging Remaining Members
After placing A, B, and the 3 chosen members, we have 7 members left. These 7 members can be arranged in:
\[ \text{Arrangements of remaining members} = 7! \]

**Hint:** Treat the remaining members as independent units that can be arranged freely.

### Step 7: Total Favorable Arrangements
Now we can calculate the total number of favorable arrangements where A and B have exactly 3 members between them:
\[ \text{Favorable arrangements} = 2 \times \binom{10}{3} \times 3! \times 7! \]

### Step 8: Calculating the Probability
The probability \( P \) that there are exactly 3 members between A and B is given by the ratio of favorable arrangements to total arrangements:
\[ P = \frac{\text{Favorable arrangements}}{\text{Total arrangements}} = \frac{2 \times \binom{10}{3} \times 3! \times 7!}{11!} \]

### Step 9: Simplifying the Probability
Now, substituting \( \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} \):
\[ P = \frac{2 \times \frac{10 \times 9 \times 8}{6} \times 6 \times 7!}{11 \times 10!} \]
This simplifies to:
\[ P = \frac{2 \times 10 \times 9 \times 8 \times 7!}{11 \times 10!} \]
Since \( 10! = 10 \times 9 \times 8 \times 7! \), we can cancel \( 7! \):
\[ P = \frac{2 \times 10 \times 9 \times 8}{11 \times 10 \times 9 \times 8} = \frac{2}{11} \]

### Final Answer
Thus, the probability that there are exactly 3 members between A and B is:
\[ \boxed{\frac{2}{11}} \]