There are 10 people lining up for a movie including three friends Anne, Bradley and Carol.

Question

a. Determine the number of ways the line could be formed if the three friends will always stand together.

b. Determine the number of ways the line could be formed if the three friends stand together and Anne and Carol are always standing next to each other.

Answer 1

Let m be the number of ways the three friends could stand with respect to each other.

In question a., there is no constraint on their permutation, so

m = 3! = 6

In question b., either Anne is before Carol or the other way round;

for each of those 2 ways, Bradley can stand in front or behind those two. So

m = 4

The other 7 people can be arranged in 7! = 5040 ways.

For each of those there are 8 places where the three friends can be inserted.

So the final result is

m×5040×8