10 persons are seated at a round table. The probability that two particular persons sit together is :

Question

Accepted Answer

To determine the probability that two particular persons sit together at a round table with 10 persons, we need to consider the arrangement of the people and the nature of circular permutations.

### Explanation for Option 2: $\frac{2}{9}$

1. **Total Arrangements in a Circle:**
   - When arranging 10 people in a circle, we fix one person and arrange the remaining 9 around them. This gives us $9!$ arrangements.

2. **Treating Two Persons as a Single Unit:**
   - If we want two particular persons to sit together, we can treat them as a single unit or "block." This effectively reduces the problem to arranging 9 units (the block plus the other 8 individuals) in a circle.
   - The number of ways to arrange these 9 units in a circle is $8!$.

3. **Arranging the Two Persons Within the Block:**
   - Within the block, the two persons can be arranged in $2!$ ways (Person A next to Person B or Person B next to Person A).

4. **Calculating the Probability:**
   - The number of favorable arrangements (where the two persons sit together) is $8! \times 2!$.
   - The total number of possible arrangements is $9!$.
   - Therefore, the probability that the two particular persons sit together is:
     $$
     \frac{8! \times 2!}{9!} = \frac{8! \times 2}{9 \times 8!} = \frac{2}{9}
     $$

### Explanation for Why Other Options are Incorrect:

1. **Option 1: $\frac{1}{5}$**
   - This option suggests a probability that is too high given the calculations. The correct probability, as derived, is $\frac{2}{9}$, which is less than $\frac{1}{5}$.

2. **Option 3: $\frac{3}{7}$**
   - This probability is not consistent with the calculations. The fraction $\frac{3}{7}$ does not match the derived probability of $\frac{2}{9}$.

3. **Option 4: $\frac{4}{9}$**
   - This option suggests a probability that is higher than the correct probability. The correct calculation shows that the probability is $\frac{2}{9}$, not $\frac{4}{9}$.

In conclusion, the correct probability that two particular persons sit together at a round table of 10 persons is indeed $\frac{2}{9}$, making Option 2 the correct choice.