Bijection means both Onto and One-One.
Now, if we have two sets with $X$ and $Y$ with $N$ elements each, each element from $Y$ has to map exactly $1$ element of $X$.
So, this simplifies your problem to place $N$ different balls to $N$ different boxes, where each box can hold only $1$ ball. In other way, you just have to find out how many ways you can arrange the $N$ elements of $Y$.
This is simple permutation i.e. $N!$.
Do Note – Bijection is only possible between two sets if both have same cardinality.