Animated Solution for Mathematics - Permutations and Combinations: The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together is

Solving the Circular Seating Problem

Welcome, future engineer. Today, we are not just solving a permutation problem; we are choreographing a social arrangement. Combinatorics is the mathematics of counting without counting, and circular permutations are where the logic gets truly elegant.

Let us dive into the problem of seating 5 girls and 7 boys at a round table such that no two girls sit together.

The Circular Constraint

Imagine you are standing in a room with 12 chairs arranged in a perfect circle. You have 7 boys and 5 girls. The condition is strict: no two girls can be adjacent.

If you try to place everyone at once, you will quickly find yourself in a chaotic web of possibilities. Instead, we use the 'Gap Method.' We treat the boys as our 'anchors.'

By placing the boys first, we create natural barriers—the gaps—that will keep the girls separated. This is the core geometric reality: the boys define the spaces, and the girls occupy them.

Anchoring the Boys

First, we seat the 7 boys. In a linear row, 7 people would arrange themselves in $7!$ ways.

But this is a round table. If we rotate the entire group, the relative seating remains the same. To eliminate this redundancy, we fix one boy in a single seat.

This breaks the rotational symmetry. Now, the remaining 6 boys can be arranged in the remaining 6 seats in $(7-1)!=6!$ ways. We have successfully established our framework: $6!$ ways to seat the boys.

The Power of the Gaps

Now, look at the spaces between the boys. If you have 7 boys in a circle, how many gaps are there?

Draw it out mentally: between boy 1 and 2, 2 and 3, ..., and finally between boy 7 and 1. There are exactly 7 gaps.

This is a beautiful property of circular arrangements: the number of gaps equals the number of objects. We have 5 girls who need to be seated, and we have 7 available gaps. To ensure no two girls sit together, each girl must occupy a unique gap.

Selecting and Arranging

We have 7 gaps and 5 girls. First, we must choose which 5 gaps the girls will occupy.

The number of ways to choose 5 gaps out of 7 is given by the combination formula:

Once the gaps are chosen, the girls are distinct individuals, so they can be arranged in those 5 chosen gaps in $5!$ ways. Therefore, the total number of ways to seat the girls is:

The Final Synthesis

To find the total number of arrangements, we multiply the ways to seat the boys by the ways to seat the girls:

Now, let us simplify. We know that:

So, our expression becomes $6!\times 21\times 5!$. We can rewrite $6!$ as $6\times 5!$. Substituting this in, we get:

Grouping the constants and the factorials, we have:

And there it is—the elegance of the final result. You have successfully navigated the constraints, respected the circular symmetry, and arrived at the solution of $126(5!{)}^2$. Keep this logical flow in your toolkit; it will serve you well in the most challenging JEE problems.