In how many ways?

Question

A university student is selecting courses for his next semester. He can choose from 3 humanities courses and 10science courses. In how many ways can he choose 7courses if fewer than 2 must be humanities courses?

Answer 1

A university student is selecting courses for his next semester. He can choose from 3 humanities courses and 10science courses. In how many ways can he choose 7courses if fewer than 2 must be humanities courses?

These questions are sort of fun as long as you have the time to think about them. "Fewer than 2" humanities courses means either 0 or 1 humanities courses the student can register for as part of their 7 courses. So, I would first break it down as follows:

a. How many ways can a student choose 0 humanities courses from 3 humanities courses and 7 science courses from 10 science courses, and

b. How many ways can a student choose 1 humanities courses from 3 humanities courses and 6 science courses f rom 10 science courses.

For a., this is a product of the two combinations "three choose zero" and "ten choose seven" =

3!/(0!·3!) = 6/(1·6) = 1 and 10!/(7!*3!) = 10·9·8/(3·2·1) =120. So, 1·120 =120 ways to choose 0 humanities courses out of 3 and 7 science courses out of 10.

For b. this is a product of the two combinations "three choose one" and "ten choose six" =

3!/(1!·2!) =3 and 10!(/6!4!) =210. So, 3*210=630 ways to choose 1 humanities courses from 3 and 6 science courses from 10.

Now, since they want to know the number of way to choose 0 humanities courses from 3 and 7 science courses from 10 OR 1 humanities course from 3 and 6 science courses from 10, we add the answers from a. and b. together to get: 120 +630 = 750