Answer:
Since it takes two to shake hands, no odd number of people can handshake, since one must be left out. Therefore 99 people cannot shake hands. Unless "handshake" means something unusual here, which seems unlikely, as it is the only handshaking activity where you can count the handshakes.
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What utter nonsense! 3 people: A, B and C can shake hands. AB shake hands, AC shake hands and BC shake hands. And the last time I looked 3 was still classified as an odd number.
The correct answer is 99*98/2 = 4851