How to prove that $n! \mid 2^n \prod_{i=1}^{n}(2i-1)$? i would like to prove that
$$
n! \mid 2^n \prod_{i=1}^{n}(2i-1)
$$
Thanks! 

HINT: First, 
$$(2n)!=\left(\prod_{k=1}^n2k\right)\left(\prod_{k=1}^n(2k-1)\right)=2^nn!\prod_{k=1}^n(2k-1)\;.$$
Now combine this with the fact that $\dbinom{2n}n$ is an integer.
