Indicate which rules you are using and which kind of enumera

Indicate which rules you are using, and which kind of enumeration problem(s) occur. In
how many ways can 6 girls and 3 boys sit on a row of 9 chairs in such a way that no two
boys sit next to each other?

Solution

There are totally 9 persons including 6 girls and 3 boys.

Since two boys cannot sit next to each other first we can make 6 girls sit in a row with gaps.

6 girls can sit in 6! ways = 720 ways

There will be 7 gaps between girls including the extreme two corners.

The three boys can be asked to sit in the 7 gaps in 7P3 ways =7x6x5= 210

So total no of ways to make them sit so that no two boys sit together = 720 x210 =151200