Originally Posted by
HallsofIvy A product of even numbers can be reduced as:
$\displaystyle 2(4)(6)\cdot\cdot\cdot(2n-2)(2n)= [(1)(2)][(2)(2)][(3)(2)]\cdot\cdot\cdot[(n-1)(2)][(n)(2)]$$\displaystyle = n!2^n$.
A product of odd numbers is more complicated- you have to put the even numbers in:
$\displaystyle (1)(3)(5)\cdot\cdot\cdot (2n-1)(2n+1)= \frac{(1)(2)(3)(4)(5)\cdot\cdot\cdot(2n-1)(2n)(2n+1)}{2(4)(6)\cdot\cdot\cdot(2n-2)(2n)}= \frac{(2n+1)!}{2^n n!}$.