JEE Mathematics 2023 Question with Solution

Given: $5$ girls and $7$ boys are to be seated around a round table.

Find: The number of circular arrangements such that no two girls sit together.

First, seat the $7$ boys around the round table. Since circular permutations are counted up to rotation, the number of ways is

After seating the boys, there are $7$ gaps between consecutive boys around the circle.

To ensure that no two girls sit together, choose $5$ of these $7$ gaps and arrange the $5$ girls in them:

Therefore, the total number of required arrangements is

Now,

Therefore, the number of ways is $126(5!)^2$. The correct option is D.

Given: We need circular seating of $12$ people with the condition that no two girls sit together.

Find: The number of valid arrangements.

A useful way in circular arrangement problems is to fix rotational symmetry by arranging one category first.

Seat the $7$ boys around the circle:

Now there are exactly $7$ available gaps around them. In order that no two girls sit together, at most one girl can be placed in each gap.

So we select $5$ gaps out of $7$, and then permute the $5$ girls:

Multiplying with the arrangement of boys,

Hence, the required number of ways is $126(5!)^2$.