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at a party, 25 guests mingle and shake hands. prove that at least one guest must have shaken hands with an even number of guests.

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Might help: The Handshaking theorem: – Sujaan Kunalan Jun 20 '13 at 3:49
up vote 5 down vote accepted

HINT: Number the guests $1$ through $25$. Let $a_k$ be the number of guests with whom guest $k$ shook hands. Suppose that the numbers $a_1,a_2,\dots,a_{25}$ are all odd.

  • Is $\sum_{k=1}^{25}a_k$ odd, or is it even?
  • Each handshake involves two people, so each handshake is counted twice in $\sum_{k=1}^{25}a_k$, once for each of the two people involved. What does this imply about whether $\sum_{k=1}^{25}a_k$ is odd or even?
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