Find a formula for: $$ \binom n0 + 2\binom n1 + 4\binom n2 + 8\binom n3 + \cdots + 2^{n-1}\binom n{n-1} + 2^n\binom nn $$ using a counting argument.

I just used the binomial t. to show it's $3^n$ but I can't find a combinatorial way. Any help?


1 Answer 1


Consider counting functions $f: \{1,2,\dots,n\} \to \{1,2,3\}$ which we know there are $3^n$ of. The binomial coefficient $\binom{n}{i}$ chooses elements not to send to $3$ and the $2^i$ counts the ways to sending these elements into $\{1,2\}$. This defines a function just by sending everything not chosen to 3.

Best, John

  • $\begingroup$ What a great argument! +1 :) $\endgroup$ Commented Jun 8, 2014 at 3:38

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