Manipulating a factorial equation

Question

How do I convert $\dfrac{1\cdot2\cdot3\cdots2n}{n!}$ to $(n+1)\cdot(n+2)\cdot\cdots\cdot(2n)$?

I don't understand how the $n + x$ terms have appeared in this equation.

Answer 1

Note that we can simplify as follow

$$\frac{(2n)!}{n!}=\frac{\color{red}{1\cdot 2 \cdot ... \cdot n} \cdot (n+1) \cdot ... \cdot 2n}{\color{red}{1\cdot 2 \cdot ... \cdot n}}= (n+1) \cdot ... \cdot 2n$$

Answer 2

You can write : $$\begin{align} 1\cdot 2\cdot 3\cdots (2n) &= (1\cdot 3\cdot 5\cdots (2n-1))\cdot(2\cdot 4\cdot 6\cdots (2n))\\ &= (1\cdot 3\cdot 5\cdots (2n-1))\cdot 2^n\cdot(1\cdot 2\cdot 3\cdots n)\\ &= (1\cdot 3\cdot 5\cdots (2n-1))\cdot 2^n\cdot n! \end{align}$$

Note that this goes beyond the question, as it shows that you can rewrite the product in a certain ways which suits your question situation. The "other" answer better suits the need of the problem.