Math 2.0

Combinatorics Theorem

Sum of binomial coefficients

Builds on Binomial coefficient

Statement

Let nn be a natural number. Then ∑k=0n(nk)=2n.\sum_{k=0}^{n} \binom{n}{k} = 2^n.

Remarks

For n=4n = 4: 1+4+6+4+1=161 + 4 + 6 + 4 + 1 = 16. The identity is the binomial theorem with x=y=1x = y = 1. Combinatorially, both sides count the subsets of an nn-element set. The number of subsets of a finite set is 2n2^n, and (nk)\binom{n}{k} of them have size kk.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Binomial_coefficient, Sums of the binomial coefficients
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