Question

5 girls and 2 boys are arranged in a straight line for several photographs to be taken. How many different arrangement are possible if (a)there are no restrictions(b) the boys must be separated?

Answer 1

It depends:
If the children are distinguishable within gender a) `5040`; b) `3600`

If the children are not distinguishable within gender a) `21`; b) `15`

Explanation:

If the children are distinguishable by gender
(e.g. switching the positions of 2 girls is considered a different arrangement)
a) gender is irrelevant (all children are different); there are `7` children and `7!= 5040` different arrangements (`7` choices of position 1 `xx 6` choices for poistion 2 `xx 5` choices for position 3 ...etc.)

b) If the boys stand together there are really only `6` places where the 2 boys can stand (we can think of this as them only taking up 1 position). Within each of these possibilities the boys could be arrange in 2 orders. So there will be #6!xx2=720xx2 = 1440 arrangements with the 2 boys standing together.

This leaves `5040-1440 = 3600` arrangements with the boys not standing together.

If the children are NOT distinguishable by gender
a) Among the `7` possible positions `2` must be chosen into which the `2` boys will be placed.
So there are `7C_2=(7!)/((8-2)!2!) = (7xx6)/2= 21` arrangements.

b) If the boys stand together, they can be treated as a single entity within a row of `6` positions with `6` possible arrangements.
Therefore the number of arrangements with them not standing together is `21- 6 = 15`