Trouble Understanding a Combinatorics Problem

Question

This question appeared on my combinatorics exam. I did not even understand the question.

Determine the number of functions, $f:\{1,2,3\} \to \{1,2,3\}$, that satisfy $$f(1)+f(3)\equiv0\ (\text{mod }2).$$

Answer 1

We would like to know how many functions from $\{ 1, 2, 3\}$ to $\{1,2,3\}$ there are for which $f(1)+f(3)\equiv 0 \mod 2$. In other words, we would like to know how many ways we can match the parity of $f(1)$ with $f(3)$.

If $f(1) = 1$ then $f(3) = 1$ or $f(3)=3$.

If $f(1) = 2$ then $f(3) = 2$.

If $f(1) = 3$ then $f(3) = 1$ or $f(3)= 3$.

In each of these cases $f(2)$ can be any of $1$, $2$ or $3$. This means there are $2\times 3$ functions satisfying $f(1)=1$, $1\times 3$ functions satisfying $f(1)=2$, and $2\times 3$ functions satisfying $f(1)=3$.

This gives $15$ total functions satisfying your criteria.