Question

Five boys and three girls are sitting in a row of eight seats. In how many ways can they be seated so that not all the girls sit side-by-side?

Answer 1

The above scenario is identical to the statement: arrange $8$ people in $8$ seats, remove scenarios where $3$ girls are sitting side by side.

To arrange $8$ people in $8$ seats is simple, it's $8!$

The scenarios where $3$ girls sit together, we can imagine them as one single entity, and that entity has $3$ different arrangements. In other words, this scenario is also identical to the product of arranging $6$ people in $6$ different seats which is equal to $6!$ and the arrangements of the $3$ girls is $3!$

Combining the two statements, we get:

Number of ways can they be seated so that not all the girls sit side-by-side = $8!-(6!3!)$ $=36000$