Question

In how many ways can $5$ boys and $5$ girls sit in a circle so that no two boys sit together?

Answer 1

Calculate the required number of ways:

It is given that the number of boys $=5$

And, the number of girls $=5$

Now, the number of ways $5$ boys can sit around the circular table $=\left(5-1\right)!$

$\Rightarrow$ the number of ways $5$ boys can sit around the circular table $=4!$

Since the boys and the girls have to sit in a circle so that no two boys sit together.

Therefore, the boys and the girls will sit in alternate positions around a circle.

So, the number of ways girls can sit in the places between two boys $=5!$

Thus, the total number of ways $=4!\times 5!$

$\Rightarrow$ the total number of ways $=\left(4\times 3\times 2\times 1\right)\times \left(5\times 4\times 3\times 2\times 1\right)$

$\Rightarrow$ the total number of ways $=24\times 120$

$\Rightarrow$ the total number of ways $=2880$

Hence, there are $2880$ ways in which $5$ boys and $5$ girls cab sit in a circle so that no two boys sit together.