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Here are two proofs.
First Proof. Notation: is a binomial coefficient, the number of ways to choose n objects taken m at a time.
which is an even integer.
Second Proof: is the number of combinations of 2m objects taken m at a time. If we have selected m objects, there are m objects left over (un-selected); the unselected objects form another combination of m objects taken from the original 2m. So we can pair every combination of size m uniquely with another (and different) combination of size m. Therefore the total number of combinations is even.