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March 28th, 2018, 02:28 PM  #1 
Newbie Joined: May 2017 From: Neverland Posts: 8 Thanks: 0  why 2n above n (binomial) is, equal or less than, 4^n ?
i mean i know that 4^n = (1+1)^(2n) and 2n above n (binomial) is equals to 2n!/n!(2nn)! but how do i put those in words to prove the equation (Title) ...? sorry i haven't seen any method in the formulas of how to write the binomial expression ... 
March 28th, 2018, 03:19 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,430 Thanks: 1315 
$\begin{align*} &4^n = \\ \\ &(1+1)^{2n} = \\ \\ &\sum \limits_{k=0}^{2n}~\dbinom{2n}{k}(1)^k(1)^{2nk} = \\ \\ &\sum \limits_{k=0}^{2n}~\dbinom{2n}{k} = \\ \\ &\dbinom{2n}{n} + \sum \limits_{k=0,~k\neq n}^{2n}~\dbinom{2n}{k} \geq \\ \\ &\dbinom{2n}{n} \end{align*}$ 

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binomial, equal, prove 
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