Question

In how many ways can 3 girls and 3 boys line up so that no 2 of the 3 boys are next to each other?

Answer 1

Split it into two series tasks (series tasks are multiplied to get the full task)
1) assign boys spots
2) assign girls into vacant spots

Answer 2

You probably will need to split up 1) further because it is not trivial.

Answer 3

1) can be split up into
1a) assign the first boy
1b) assign the second boy
1c) assign the second boy

Answer 4

1a) is easy, there are 6 places to put him

Answer 5

For 1b), it depends on 1a), if he was put on an edge, we eliminate one place. otherwise we eliminate two places

Answer 6

1c) is like 1b) but there are even cases.

Answer 7

even more cases^ gets complicated.

Answer 8

It might actually be easier to count how many ways you CAN'T assign the boys and subtract that to the total number of assignments

Answer 9

There are 3! permutations of the boys and 3! of the girls considered separately. Each of the permutations of boys must be interleaved with each of the permutations of girls such that we get either BGBGBG or GBGBGB.
Therefore there are actually 2 * 3! permutations of boys when the options of boy first or girl first are considered. Similarly there are 2 * 3! permutations of girls.
The total number of permutations when girls and boys are interleaved is:
2 * 3! * 2 * 3! = 144