Math Help!?

Question

Suppose you are managing 15 employees, and you need to form three teams to work on three different projects.

Assume that all employees will work on exactly one team. Also, each employee has the same qualifications/skills, so any employee can serve on any team.

How large is the sample space for all ways the three teams be chosen when the number of employees on each project are as follows:

6 on Team 1

4 on Team 2

5 on Team 3

Answer 1

How would you say it in words? "Of the 15, choose 6 for Team 1; of the remaining 9, choose 4 for Team 2; of the remaining 5, choose 5 for Team 3."

(Okay, so you probably wouldn't say "of the remaining 5, choose 5")

How do you write that in terms of combinations?

(15 choose 6)*(9 choose 4)*(5 choose 5)

Suppose you chose the Team 2 members first, then the Team 3 members? Then it would be

(15 choose 4)*(11 choose 5)*(6 choose 6)

which had better give the same answer as the first calculation.

You will find that all the possibilities can be written as

15!/(4! 5! 6!)

This is the size of the sample space.