X and Y stand in a line with 6 other people. What is the probability that there are 3 persons between them?(a) 1/5(b) 1/6(c) 1/7© The Institute of Chartered Accountants of India14(d) 1/3

To find the probability that there are 3 persons between X and Y in a line with 6 other people, we need to consider the total number of possible arrangements.

First, let's calculate the total number of arrangements. Since there are 8 people in total (X, Y, and the 6 others), we can arrange them in 8! (8 factorial) ways.

Next, let's calculate the number of arrangements where there are 3 persons between X and Y. We can think of X, Y, and the 3 persons between them as a group. So, we have 5 entities to arrange: the group of 3 persons and the remaining 3 people. The group of 3 persons can be arranged in 3! ways, and the remaining 3 people can be arranged in 3! ways as well. Therefore, the number of arrangements with 3 persons between X and Y is 3! * 3!.

Finally, we can calculate the probability by dividing the number of favorable arrangements (3! * 3!) by the total number of arrangements (8!). So, the probability is (3! * 3!) / 8!.

Now, let's simplify the expression. 3! is equal to 3 * 2 * 1, which is 6. So, the probability is (6 * 6) / 8!.

To further simplify, we can calculate 8! as 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1, which is 40320. Therefore, the probability is (6 * 6) / 40320.

Simplifying further, we get 36 / 40320, which can be reduced to 1 / 1120.

So, the probability that there are 3 persons between X and Y is 1 / 1120.

Therefore, the correct answer is (c) 1/7.