TITLE: Let F be a set of $1$-to-$1$ functions from the set $\{1,2,....,n\}$ to the set $\{1,2,....m\}$ where $m \geq n \geq 1$?
QUESTION [6 upvotes]: How many functions are members of $F$?
I was asked a question like this.
I've given the answer 
$$\dfrac{m!}{(m - n)!}$$
But they said i was wrong and the answer is $mn$
Where am i wrong?
Please help. Thanks !

REPLY [2 votes]: We have $m$ choices for what $1$ is sent to. For each of these, we have $m-1$ choices for what $2$ is sent to. For each way of doing these two things, there are $m-2$ choices for what $3$ is sent to, and so on for a total of
$$m(m-1)(m-2)\cdots(m-n+1).$$
This can also be written as $\frac{m!}{(m-n)!}$. 
Your calculation is correct. The answer $mn$ is not, unless the problem was incorrectly described.