The number of ways of choosing 2 prefects from 5 is

The number of ways of choosing 6 non-prefects from 10 is

(a) Number of possible committees with exactly 2 prefects:

(b) Number of committees with 3 prefects:

Number of committees with 4 prefects:

Number of committees with 5 prefects:

So the number of committees with at least 2 prefects is:

2100 + 2520 + 1050 + 120 = 5790

Alternative Solution:

The problem with the method used above is that if we have many (say 20) to count, it would become very tedious. So we look at another way of doing it.

If we find the number of committees with 0 prefects and 1 prefect, and subtract this from the total number of committees, we will have the number with at least 2:

Number of committees with 0 prefects:

Number of committees with 1 prefect:

The total number of committees is:

So the number with at least 2 prefects is given by:

6435 − 45 − 600 = 5790