Exercise 404Combinatorics › Binomial coefficientsHigh schoolAdded 3 Nov 2026

Sum of binomial coefficients

#sum identity#2^n#binomial theorem

Statement

Show that $\sum_{k=0}^{n}\binom{n}{k}=2^n$ .

Hint

Set $a=b=1$ in the binomial theorem.

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Binomial theorem. $(1+1)^n=\sum_k\binom{n}{k}1^{n-k}1^k=\sum_k\binom{n}{k}$ .

Conclusion. $\sum_{k=0}^{n}\binom{n}{k}=2^n$ .

Combinatorial interpretation. $2^n$ counts all subsets of an $n$ -set; $\binom{n}{k}$ counts those of size $k$ .