Each of the ten shakes nine hands, which makes 90, but that counts every handshake twice, so it’s 45. Or count them off: the first person shakes 9 hands, the next 8 new ones, and so on down to 1.
Ten people at a party each shake hands once with every other person there. How many handshakes are there?
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Every handshake takes two people. Don’t count any twice.
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45
Source: Traditional