Combinatorics Problem: Employee grouping problem

Question

Problem

A company employs eight people in the marketing department, five in the manufacturing department and three in the financing department. A project team of six is to be formed. In how many ways can the team to be formed if:

1. there are to be two representatives from each department?
2. there are at least two members from the marketing department?

My Attempt

For the first question, I was able to do it:

$$5 × 4 × 24 × 23 = 11040.$$

However, I am unable to get the second question. I tried $$16 × 15 × 14 × 13 × 12 × 11$$ but that is wrong.

What am I doing wrong? How would I go about solving these kinds of problems.

Answer 1

No restrictions: $\displaystyle \binom{16}{6}$

No member is from marketing department: $\displaystyle \binom{8}{6}$

Exactly one member is from marketing department: $\displaystyle \binom{8}{1}\binom{8}{5}$

At least two members are from marketing department: $\displaystyle \binom{16}{6}-\binom{8}{6}-\binom{8}{1}\binom{8}{5}=7532$

Answer 2

In how many ways can the team be formed if there are two members from each department?

Your answer is incorrect.

We must choose two of the eight members of the marketing department, two of the five members of the manufacturing department, and two of the three members of the financing department. Therefore, the number of teams that can be formed is $$\binom{8}{2}\binom{5}{2}\binom{3}{2}$$

In how many ways can the team be formed if there are at least two members of the marketing department?

There are eight members of the marketing department and $5 + 3 = 8$ other employees available. We subtract those teams with fewer than two members from the marketing department from the total. If there were no restrictions, we could form $\binom{8 + 5 + 3}{6} = \binom{16}{6}$ teams of six people. Of these, $\binom{8}{0}\binom{5 + 3}{6} = \binom{8}{6}$ contain no members of the marketing department and $\binom{8}{1}\binom{5 + 3}{5} = \binom{8}{1}\binom{8}{5}$ contain exactly one member of the marketing department. Therefore, the number of teams of six employees that contain at least two members of the marketing department is $$\binom{16}{6} - \binom{8}{6} - \binom{8}{1}\binom{8}{5}$$