Although graph theory is sometimes included in an undergraduate combinatorics course, we choose to exclude it. Graph theory is important enough to be studied on its own.

At a party the number of guests who shook hands an odd number of times is even.

\(\displaystyle n\binom{n}{n} + (n - 1)\binom{n}{n - 1} + \ldots + 0\binom{n}{0} = n2^{n - 1}\)

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\(\displaystyle \binom{p + q}{k} = \binom{p}{0} \cdot \binom{q}{k} + \binom{p}{1} \cdot \binom{q}{k - 1} + \ldots\)