Algebra II: Word Problem #1

Question

There are five main characters in a screenplay, and the writer wants to do scenes with every pair. The writer also wants to do scenes with every group of three characters, as well as scenes with every group of four characters. How many scenes is this movie going to have?

Answer 1

If you mean 1 scene for every group, you want ₅C_{2 + 5}C₃ + ₅C₄ ... where _nC_r is the binomial coefficient of from n pick r.  However, the groups with 3 will also have the same people as some of the groups of 2, but if that is irrelevant to our question, then my initial answer is correct.