Sum of binoms

( 18 0 ) + ( 18 1 ) + ( 18 2 ) + + ( 18 18 ) = ? \large \binom{18}{0}+\binom{18}{1}+\binom{18}{2}+\dots{}+\binom{18}{18}=?


The answer is 262144.

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1 solution

Matin Naseri
Mar 10, 2018

For ( n 0 ) + \binom{n}{0}+ ( n 1 ) + \binom{n}{1}+ ( n 2 ) + \binom{n}{2}+ + \dots{}+ ( n n ) = \binom{n}{n}= we have 2 n \text{2}^{n}

Thus for ( 18 0 ) + \binom{18}{0}+ ( 18 1 ) + \binom{18}{1}+ ( 18 2 ) + \binom{18}{2}+ + \dots{}+ ( 18 18 ) = x \binom{18}{18}=x

x = 2 18 = 262144 \text{}x={2}^{18}=262144

\therefore the answer is 262144 \boxed{\color{#302B94}{262144}}

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