If $x$ and $y$ are positive integers then $\frac{(2x)!(2y)!}{x!y!(x+y)!}$ is an integer
I have to show that the proposition above is true for any $x,y\in\mathbb{Z^+}$ by means of Legendre's formula..this is what I have:
If $p$ is a prime number, then in $m!$ there are $\left\lfloor{\frac{m}{p^n}}\right\rfloor$ multiples of $p^n$, so it suffices to prove that $$\left\lfloor{\frac{x}{p^n}}\right\rfloor+\left\lfloor{\frac{y}{p^n}}\right\rfloor+\left\lfloor{\frac{x+y}{p^n}}\right\rfloor\leq \left\lfloor{\frac{2x}{p^n}}\right\rfloor+\left\lfloor{\frac{2y}{p^n}}\right\rfloor$$ and if we make $\frac{x}{p^n}=c_1$ $\frac{y}{p^n}=c_2$ then $$\left\lfloor{c_1}\right\rfloor+\left\lfloor{c_2}\right\rfloor+\left\lfloor{c_1+c_2}\right\rfloor\leq \left\lfloor{2c_1}\right\rfloor+\left\lfloor{2c_2}\right\rfloor$$ but from here I don't know how to proceed...
Any help would be really appreciated