My approach
Let $x$ be a sufficiently large number such that the product of numbers from $x$ to $x-(2n-1)$ is $P$
$$P=x(x-1)(x-2) \ldots (x-(2n-1))$$ $$P(x-2n)!= x!$$ $$P=\frac{x!}{(x-2n)!}$$
Now, we have to prove that $(2n)!$ divides $P$.
Cannot continue...
I can't take it from here. I tried proving by expanding it but failed. Any other approach to the problem is welcome.