<s>[INST] show that $(n+1)(n+2)...(2n)$ is divisible by $2^n$ but not by $2^{n+1}$ Is this proof correct?
Suppose $2^k$ is the largest power of $2$ in the sequence $n+1, n+2, ... 2n$
Then we can compute the power of 2 in the product as $n/2 + n/2^2 + ... n/2^k = n(1 + 2 + .... 2^{k-1})/2^k = n$. 
 [/INST] You can also proceed by induction.
Let $a_n = (n+1)(n+2)\ldots(2n)$.  
Then $a_{n+1}/a_n = (2n+1)(2n+2)/(n+1) = 2(2n+1)$.
Since $2n+1$ is odd, only one $2$ factor is added at each step.
Since $a_1 = 2^1$, you get $a_n = 2^n \times o_n$ where $o_n$ is an odd number 
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