12 persons are seated around a round table. What is the probability that two particular persons sit together?

Options

A. `2/11`
B. `1/6`
C. `3/11`
D. `3/15`

Answer

To find the probability that two particular persons sit together at a round table with 12 persons, we can follow these steps:

### Step 1: Calculate the total arrangements of 12 persons around a round table.
When arranging \( n \) persons in a circle, the number of arrangements is given by \( (n-1)! \). Therefore, for 12 persons:

\[ \text{Total arrangements} = (12 - 1)! = 11! \]

### Step 2: Treat the two particular persons as a single unit.
If we want the two particular persons (let's call them A and B) to sit together, we can treat them as one single unit or block. This means we now have 11 units to arrange (the block of A and B plus the other 10 persons).

### Step 3: Calculate the arrangements of these 11 units.
Now, we can arrange these 11 units in a circle:

\[ \text{Arrangements of 11 units} = (11 - 1)! = 10! \]

### Step 4: Account for the arrangements within the block.
Since A and B can switch places within their block, we have 2 arrangements for A and B (AB or BA). Therefore, we multiply the arrangements of the 11 units by the arrangements within the block:

\[ \text{Favorable arrangements} = 10! \times 2 \]

### Step 5: Calculate the probability.
The probability that A and B sit together is given by the ratio of favorable arrangements to total arrangements:

\[ \text{Probability} = \frac{\text{Favorable arrangements}}{\text{Total arrangements}} = \frac{10! \times 2}{11!} \]

### Step 6: Simplify the probability.
We know that \( 11! = 11 \times 10! \), so we can rewrite the probability as:

\[ \text{Probability} = \frac{10! \times 2}{11 \times 10!} = \frac{2}{11} \]

### Final Answer:
Thus, the probability that the two particular persons sit together is:

\[ \frac{2}{11} \]

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