counting exercises

Question

A company has 6 officers and 4 directors (10 different people). In how many ways can a committee of 4 people be chosen if the committee must have 3 officers and 1 director?

Answer 1

a company has 6 officers and 4 directors

the committee of 4 people must contain 3 officers and 1 director

how many ways can you choose 3 officers out of 6 officers ?

6!/3!(6-3)!

6!/3!(3!)

(6*5*4)/(3*2*1)=120/6=20 ways to choose 3 people out of 6 people

you have 20 groups of 3 officers

each group can have 1 of 4 different directors

20*4=80 groups consisting of 3 officers and 1 director

call the directors A,B,C, and D

each of the 20 groups can be with A (20 groups)

each of the 20 groups can be with B (20 more groups)

each of the 20 groups can be with C (20 more groups)

each of the 20 groups can be with D (20 more groups)

total again is 20*4=80 groups