# Prove that for all positive integer n, the inequality $2n\choose n$ $<4^n$ holds [duplicate]

How do I prove that for all positive integer n, the inequality $2n\choose n$$<4^n$ holds?

Thank you!

• Oct 4, 2016 at 9:13

Hint: The LHS is the number of $n$-element subsets of $[2n]$, while the RHS is the number of all subsets of $[2n]$.
• @Emile You can use this hint the following way: $4^n=(1+1)^{2n} =\sum_{k=0}^{2n} \binom{2n}{k} > \binom{2n}{n}$. May 16, 2013 at 16:32