Math 2.0

Combinatorics Theorem

Symmetry of binomial coefficients

Builds on Binomial coefficient

Statement

Let nn and kk be natural numbers with k≤nk \le n. Then (nk)=(nn−k).\binom{n}{k} = \binom{n}{n - k}.

Remarks

For a set XX with nn elements, the map S↦X∖SS \mapsto X \setminus S is a bijection from the kk-subsets of XX to the (n−k)(n-k)-subsets. This is a bijective proof of the identity. The identity makes each row of Pascal's triangle read the same from left to right and from right to left.

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Binomial coefficient, definition, no evidence levelBinomial coefficientdefinitionSymmetry of binomial coefficients, statement, no evidence levelSymmetry of binomial …statement
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  1. Binomial coefficient definition
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Further reading

  1. Web https://en.wikipedia.org/wiki/Binomial_coefficient, Factorial formula
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