Question

In how many ways 3 girls and 3 boys be seated in a row of 6 chairs if 1 boy and 1 girl refused to sit next to each other?

Answer 1

`color(blue)(480 " Ways")`

Explanation:

First we find the number of permutations of the 3 girls and 3 boys filling the 6 chairs, this is not accounting for the two who won't sit together.

`color(white)(0)^6P_6=(6!)/((6-6)!)=6xx5...2xx1=720`

For the number of ways with the two who refuse to sit together being put together:

If the said boy is in seat 1 and girl in seat 2.

There are `4!` ways for the others to fill the remaining chairs.

If the said boy is in seat 2 and girl in seat 1.

There are `4!` ways for the others to fill the remaining chairs.

If we do this for seat 2 & 3, 3 & 4 etc. We find we have:

`10 xx 4!` ways of the said boy and girl being next to each other.

We therefore need to subtract this from `6!`

`6!-4! =480`