Everyone at a party of 20 shakes hands with everyone else once. How many handshakes?
At a party of 20 people, every guest shakes hands with every other guest exactly once, and no one shakes their own hand or repeats a handshake. Without counting them one by one, is there a quick way to work out the total number of handshakes that happened?
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190 handshakes. Each of the 20 people shakes hands with the 19 others, giving 20 times 19 = 380, but that counts every handshake twice (once from each person's side), so the true total is 380 divided by 2, which is 190. In general, for n people, the total is n(n-1)/2, the same formula that generates the triangular numbers, and the identical logic explains why a league of n teams needs exactly that many matches for everyone to play everyone once.