Question

5 boys and 5 girls will sit side by side. Any 2 girls won't sit side by side. How many different ways they can sit side by side?

Answer 1

Assume the boys and girls are not identical twins, so they are distinguishable,
If we ignore the girls, the boys can be in any of 5! orders.
Now imagine placing the girls after they boys are already in place. There are six spots for girls: Before the first boy, between the first and second, etc. The fact that no two girls can be adjacent means that no two can go in the same spot. The number of ways to place the girls is therefore (6 choose 5)⋅5!.
This gives us a total of (6 choose 5)⋅5!⋅5!=86400
Do those steps all make sense?