Product representation of $(2n)!$?

Question

I know that $n!:=\prod_{k=1}^nk$. Would it just be $(2n)!:=\prod_{k=1}^n2k$ or $(2n)!:=\prod_{k=1}^{2n}k$

Any ideas?

Answer 1

The second one. The first one is $n!2^n$.

Answer 2

Let us write them out:

$$\prod_{k=1}^{n}k=1\cdot 2\cdot 3\cdot\ldots\cdot n=n!$$

$$\prod_{k=1}^{n}2k=2\cdot 4\cdot 6\cdot\ldots\cdot 2n=2^nn!$$

$$\prod_{k=1}^{2n}k=1\cdot 2\cdot 3\cdot\ldots\cdot(2n)=(2n)!$$