In any session, the number of people who shook hands an odd number of times must be even.  This can be shown by noting that a handshake always involves two people.  There must therefore be an even number of person- handshakes overall.  Since an even number multiplied by any integer is always even, it doesn’t matter how many people shook hands an even number of times.  But since odd times odd equals odd, the tallies show an odd number of person-handshakes, which is impossible.  Hence there is a mistake.