I need to prove $\sum \binom nk 2^k = 3^n$, combinatorially.
I have already proved this identity using binomial theorem, simply by substituting x = 1 and y = 2.
Could I get some help/tips on how to prove this combinatorially?
2026-03-27 16:47:14.1774630034
Prove $\sum \binom nk 2^k = 3^n$ this identity combinatorially
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To count the number of strings of length $n$ from the alphabet $\{a,b,c\}$, first choose how many positions in the string will be either $a$ or $b$. Call that number $k$. There are $\binom nk$ ways to select which $k$ positions will be $a$ or $b$ (all the other positions will be $c$), and once you have chosen a set of $k$ positions there are $2^k$ ways to choose if each of them ends up an $a$ or a $b$.