Discrete math and structures

Question

Suppose you are managing 17 employees, and you need to form three teams to work on different projects. Assume that all employees will work on a team, and that each employee has the same qualifications/skills so that everyone has the same probability of getting choosen. In how many different ways can the teams be chosen so taht the number of employees on each project are as follows:

6,3,8

Answer 1

This problem is solved using the multinomial coefficient:

Suppose we have n interchangeable objects and k groups to put them in; we put n₁, n₂, ..., n_k objects in each group, where n₁ + n₂ + ... + n_k = n. The number of ways to do this is equal to (n!) / [(n₁!) × (n₂!) × ... × (n_k!)], which is know as the multinomial coefficient.

Here, we have n = 17 employees (which are interchangeable because each employee has the same qualifications/skills) and k = 3 groups to put them in. We also have n₁ = 6, n₂ = 3, n₃ = 8 (note that 6 + 3 + 8 = 17). Therefore, the number of teams that can be chosen is (17!) / [(6!) × (3!) × (8!)], which turns out to be 2,042,040.