Finding the number of ways that five boys and three girls can be seated in a row such that not all girls sit side by side

Question

Five boys and three girls are to be seated in a row such that no two girls sit side by side?

Answer 1

I'm assuming there is individuality between the girls and the boys, so we can generalize arrangements such that there is no need to remove cases of congruent order.

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 - 6! - and the arrangements of the 3 girls, 3!

Combining the two statements, we get:

8! - (6!3!) = 36000

Answer 2

So you should first try to count how many ways can everybody be seated without any constraint. Keep in mind that we don't care about specific people, namely (boy 1 on chair 1, boy 2 on chair 2) is exactly the same as (boy 2 on chair 1, boy 1 on chair 2).

Then you can just remove from this total the number of ways you can sit all girls side by side. This is really easy to picture in your mind.

(Actually it is not mentioned whether boys and girls are supposed to be distinguishable or not, here I assumed they are not)