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Solutions_2010

2010-12-6 number theory quickie by Stumbo

Let $n, k$ be 2 positive integers with $n\geq 2k$.
Show that either $\binom{n}{k}$ or $\binom{n-k}{k}$ is divisble by 2.
Solution by Stumbo
The product of those binomial coefficients, divided by two, is the number of unordered disjoint pairs of K-subsets of an N-set. Thus the product of the coefficients is even, so at least one of them must be even.