Suppose $X$ contains m elements and $Y$ contain $n$ elements. The number of functions from $X$ to $Y$ is 

Let $X=\left\{x_1,x_2,\dots x_{\mathrm{m}}\right\}$ and $Y=\left\{y_1,y_2,\dots y_n\right\}$

For each $x_1(1\le i\le m)$ we have n possible images. Therefore, the number of functions from X to Y is $n^m$