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Concept# Bijective proof

Summary

In combinatorics, bijective proof is a proof technique for proving that two sets have equally many elements, or that the sets in two combinatorial classes have equal size, by finding a bijective function that maps one set one-to-one onto the other. This technique can be useful as a way of finding a formula for the number of elements of certain sets, by corresponding them with other sets that are easier to count. Additionally, the nature of the bijection itself often provides powerful insights into each or both of the sets.
Basic examples
Proving the symmetry of the binomial coefficients
The symmetry of the binomial coefficients states that
: {n \choose k} = {n \choose n-k}.
This means that there are exactly as many combinations of k things in a set of size n as there are combinations of n − k things in a set of size n.
A bijective proof
The key idea of the proof may be understood from a simple example: selecting

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