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 $$(2n)!$$

Hint: You can verify by a computation that
$$\frac{1}{n+1}\binom{2n}{n}=\binom{2n}{n}-\binom{2n}{n+1}.$$
Detail: We have
$$\frac{1}{n+1}\binom{2n}{n}=\frac{1}{n(n+1)}\frac{(2n)!}{(n-1)!n!}=\left(\frac{1}{n}-\frac{1}{n+1}\right) \frac{(2n)!}{(n-1)!n!}    .$$
Now
$$\frac{1}{n}\frac{(2n)!}{(n-1)!n!} =\binom{2n}{n}\qquad\text{and}\qquad \frac{1}{n+1}\frac{(2n)!}{(n-1)!n!} =\binom{2n}{n+1}              .$$
