EXAMPLE 1.6.6

The mathematics department is going to hire a new instructor. They want to hire somebody who possesses at least four of the following traits:

1. Honest;

2. Trustworthy;

3. Loyal;

4. Gets along well with others;

5. Good at math;

6. Good handwriting

In how many ways is it possible to combine at least four of these traits?

EXAMPLE 1.6.6 SOLUTION

In this case, "at least four traits" means "either 4 traits or 5 traits or 6 traits." We must find the number of ways to combine 4 traits, then find the number of ways to combine 5 traits, then find find the number of ways to combine 6 traits, and ADD these three numbers.

Since there are 6 traits from which to choose, the number of ways to combine (choose) exactly 4 of them is C(6,4).

There are 15 ways to choose exactly 4 of the 6 traits.

The number of ways to choose exactly 5 traits is C(6,5).

There are 6 ways to choose exactly 5 of the 6 traits.

The number of ways to choose exactly 6 of the traits is C(6,6).

There is one way to choose exactly 6 of the 6 traits.

So, the number of ways to choose either 4 traits or 5 traits or 6 traits is 15 + 6 + 1 = 22.