Proof by storytelling

There are many equivalent ways of defining the binomial coefficients $\binom{n}{r}$ (pronounced ‘$n$ choose $r$’).  In this article, though, $\binom{n}{r}$ is defined simply as the number of ways of choosing a subset of $r$ things from a set of $n$ things.  Note that this definition does not give us a way to calculate $\binom{n}{r}$; and, if you already know how to evaluate the binomial coefficients, you should put the formula out of your mind and let yourself be surprised by the beautiful way in which identities can be proved without resorting to bashing out messy fractions of factorials.

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