What is the meaning of $(2n)!$ I came across something that confused me $$(2n)!=?$$ What does this mean: $$2!n!, \quad  2(n!)$$ or $$(2n)!=(2n)(2n-1)(2n-2)...n...(n-1)(n-2)...1$$ 
Which one is right? 
The exercise is to show that $$(n+1)\bigg|\left(\begin{array}{c}2n\\n\end{array}\right)$$Then I thought of using the combination formula $\left(\begin{array}{c}n\\k\end{array}\right)=\frac{n!}{k!(n-k)!}$ to decrease my expression, but then I came across $$(2n)!$$

You are correct:
$$(2n)! = (2n)(2n-1)(2n-2)\cdots(3)(2)(1)$$
So, for example, if $n= 3$, then $(2n)! = 720$.
