Question

A committee of 5 is chosen from 6 teachers and 3 students and 7 others. Determine the number of ways of selecting the committee if it is to contain: (a)at least two teachers (b)no teacher nor student

Answer 1

a) An efficient method is to consider the number of ways to include 0 or 1 teachers, then subtract that from the total number of ways to make a committee.
0 teachers: 10c5
Exactly one teacher: 6c1*10c4
Total ways to choose a committee with no restrictions: 16c5
Thus, the total ways to choose a committee with at least 2 teachers is:
16c5 - 10c5 - 6c1*10c4 = ______

b) no teachers, no students: 7c5 = ______