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? (source: New Revised GRE General Guide)
I initially tried to solve the problem by doing: $\binom{4}{2} \binom{8}{2}$ (i.e., (4 choose 2) * (8 choose 2)). That gives me 168.
However, the answer is actually 115. I understand how the answer solved it: look at 3 cases:
- Case 1: 2 bios and 2 fiction
- Case 2: 3 bios and 1 fiction
- Case 3: 4 bios and 0 fiction
And add up the number of combinations of each.
But what's wrong with my approach?