From a group of 10 people, 5 are assigned to room A and two to room B. In how many ways can the assignment be

Question

From a group of 10 people, 5 are assigned to room A and two to room B. In how many ways can the assignment be

Answer 1

2,520

First, group A: (10 x 9 x 8 x 7 x 6)/(5 x 4 x 3 x 2 x 1) = 252

Then group B: (5 x 4)/(2 x 1) = 10

Or first group B: (10 x 9)/(2 x 1) = 45

Then group A: (8 x 7 x 6 x 5 x 4)/(5 x 4 x 3 x 2 x 1) = 56

252 x 10 = 45 x 56 = 2,520

Another approach:

The general formula: The number of groups of size i that can be selected from a group of size n, where i<=n, is n!/(n-i)!i!.

So group A, i = 5, n = 10, so 10!/(10-5)!5! = 3,628,800 / (120 x 120) = 252.

Group B, i = 2, n = 5, so 5!/(5-2)!2! = 120 / 6 x 2 = 10.

252 x 10 = 2,520.

Answer 2

30