Handshaking lemma

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I'm collecting mathematical facts that can be easily explained to non-mathematicians and that have both "unimportant" and very important applications. For example, Theorema Egregium can be applied to eating pizza and is very important in creating maps.
Handshaking lemma has an obvious "application" to counting handshakes at a party. It is also very useful in proofs and in general graph theory. I can't think of a concrete important example though, easy to explain within a short time.
Any ideas about handshaking lemma or similar examples would be appreciated.