Intuition behind combination problem

Question

I came across this question on a GRE practice exam. I'm trying to build intuition behind how to correctly approach these types of problems:

A reading list for a humanities course consists of 10 books, of which 4 are biographies and the rest are novels. Each student is required to read a selection of 4 books from the list, including 2 or more biographies. How many selections of 4 books satisfy the requirements?

Initially I thought you could solve the problem by finding out 4C2 (meaning choose 2 out of 4) and then group the remaining two biographies with the novels, leaving 8 books to choose from. Then I would solve for 8C2 to find out how many ways I could select the other 2 books from the remaining 8 and multiply that with the result from 4C2. Can somebody explain why my reasoning is wrong?

Thank you.

Answer 1

Your method double-counts. You could first choose biographies 1 and 2, and then from the remaining 8 books choose biographies 3 and 4. Or you could first choose biographies 3 and 4, and then from the remaining 8 choose biographies 1 and 2. Your method counts these as two different selections, but of course they're the same.

I would approach this by adding together:

• the number of ways to choose the 4 books such that exactly 2 are biographies
• the number of ways to choose the 4 books such that exactly 3 are biographies
• the number of ways to choose the 4 books such that exactly 4 are biographies

Answer 2

I would count the size of the complement, i.e., compute how many selections do not satisfy the criteria. That is, count the number of selections that include 0 or 1 biographies, and then subtract that from the total number of selections, which is $\binom{10}{4}$.