TITLE: Some inequalities on Binomial Coefficient (Question from NBHM, India)
QUESTION [0 upvotes]: Let $n\in N. $ Which of the following inequalities are TRUE.
$(a)$ For every $n>1$, $ \quad  {{2n}\choose{n}}^{1/n}> 2.$
$(b)$ For every $n\geq1$ , $\quad {{2n}\choose{n}}< \frac{1}{\sqrt{2n+1}}.$
$(c)$For every $n>1$,$ \quad  n!{{2n}\choose{n}}< (2n)^n.$
Kindly help I have no idea how to solve these type of combinatorial inequalities. Thank you.

REPLY [0 votes]: Mostly, cleared by Stella Biderman, I just explain option (a) By induction: Let ${{2k}\choose{k}}\geq 2^k$ holds for all $k\leq n$, then ${{2n+2}\choose{n+1}}=\frac{(2n+2)(2n+1)}{(n+1)^2}{{2n}\choose{n}}\geq \frac{(2n+2)(2n+1)}{(n+1)^2}2^n= \frac{(2n+1)}{(n+1)} 2^{n+1}\geq2^{n+1}$
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