Question

In a class, there are 10 boys and 8 girls.

(a) In how many ways can the teacher arrange 6 boys and 4 girls in a row if the girls must stand in the middle of the row?

(b) In how many ways can the teacher arrange 6 boys and 4 girls in a row if no girls stand next to another girl?

Answer 1

Answer)

A)

As we need girls in center, we will consider all the girls as a 1 unit.

So now we need to arrange 6 boys and 1 girl

There should be 3 boys left to the girls and 3 boys right to the girl.

Those 3 boys in right and left can be arranged in 3! Ways and girls can be arranged in 4! Ways.

So the answer is 3!*4!*3! = 864

B)

No girl should be next to each other.

6 boys can be arranged in 6! Ways.

And girls now have 7 places as they cannot be placed next to each other.

But we have to arrange 4 girls

7c4 = 7!/(4!(7-4)!)

And those 4 girls can be arranged in 4! Ways.

Answer is

6!*[(7!/(4!(7-4)!)]*4!

= 604800