Probability, arrangements

Question

A family of 7 are to take a photograph.

a) How many arrangements are possible if the triplets (Jack, Bob, Don) are side by side

b) How many arrangements are possible if the Parents (Alex and Molly) are not side by side

Answer 1

a) X is the triplets, _ is anyone else

XXX _ _ _ _

_ XXX _ _ _

_ _ XXX _ _

_ _ _ XXX _

_ _ _ _ XXX

in the first scenario the 4 people can be arranged 4! ways in each scenario,

so 5 * 4! = 5*24 = 120

b) X and Y represent the parents

X _ _ _ _ _ Y

_ X _ _ _ _ Y

_ _ X _ _ _ Y

_ _ _ X _ _ Y

_ _ _ _ X _ Y

the same scenarios occur with Y and X switched...

in each scenario the 5 other people can be arranged 5! ways

so 2 *5 * 5! =

10 * 120 = 1200