Question: 2017 Fall Midterm - 4

Author: Michiel Smid

Let $m \geq 5$ and $n \geq 5$ be integers. Let $P$ be a set consisting of $m$ strictly positive integers, and let $N$ be a set consisting of $n$ strictly negative integers. Consider 5-element subsets $A$ of $P \cup N$ such that the product of the elements in $A$ is strictly positive. How many such subsets $A$ are there?

(a)

${m+n \choose 5} - {n \choose 5}$

(b)

${m \choose 5} \cdot {n \choose 5}$

(c)

${m \choose 5} + {m \choose 3} \cdot {n \choose 2} + m \cdot {n \choose 4}$

(d)

${n \choose 5} + {n \choose 3} \cdot {m \choose 2} + n \cdot {m \choose 4}$