You are foreman foreperson of a construction crew of 8 worke

You are foreman (foreperson?) of a construction crew of 8 workers. You need some of them to cut wood, some to nail in plywood boards, and some to sand the floors. If each task must have at least 2 people working on it, how many ways are there to assign people to tasks?

Solution

The sample space is

(2,2,4),(2,3,3),(2,4,2),(3,2,3),(3,3,2),(4,2,2)

The three cases that have 4 workers on some task, the number of ways to assign them is

8C2 * 6C2

The other three cases, the number of ways to assign the workers is

8C2 * 6C3

So the answer is

3*8C2 * 6C2 +3*8C2 * 6C3

=3*(8*7/2) * (6*5/2) + 3*(8*7/2) * (6*5*4/6)

=2940