I have this solution but I don't know if is right
Let be bijective functions $f\_{i}$ with i=1,2,...,k and $f\_{i}:X\mapsto Y$
Now $f(x\_{1})=y\_{1},y\_{2},..,y\_{m}$ and this take the value of m.  

But $$ f(X\_{1})\neq f(X\_{2})\neq ...\neq f(X\_{k})$$
We know that they aren´t equal so $$f(X\_{1})=m\\
f(X\_{2})=m-1\\
\vdots \\
f(X\_{k})=m-(m-1)$$
so $(m)(m-1)...(m-(m-1))$