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

I would think about this problem as follows:

Number of ways to read 2 biographies * number of ways to read 2 novels, PLUS

number of ways to read 3 biographies * number of ways to read 1 novel, PLUS

number of ways to read 4 biographies * number of ways to read 0 novels

₄C₂*₆C₂ + ₄C₃*₆C₁ + ₄C₄*₆C₀ = 90 + 24 + 1 = 115

In the method you described, you are choosing from the biographies in an invalid way (overlapping in both choose-from sets). To see the overlapping, note that you are choosing from 4 then you are choosing from 8, but you only have 10 books.

Answer 2

Your reasoning is only slightly incorrect. The issue is that you're not constrained to choose 2 biographies, but rather at least 2. This means you can also choose 3 biographies or 4. So, the number of outcomes is:

4 Biographies: 4 C 4 = 1

3 Biographies, 1 novel: 4 C 3 * 6 C 1 = 24

2 Biographies, 2 novels: 4 C 2 * 6 C 2 = 6 * 15 = 90

So, the total options are 90 + 24 + 1 = 115.

Notice how that's very close to your answer: 4 C 2 * 8 C 2 = 112.