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If nine people each shook hands with everybody else once, a total of36 handshakes would take place. Here is the easy, simple and elegant way to solve this: 9 x 8 = 72 72 ÷ 2 = 36. This is the harder way, but it might be easier to understand: The first person shakes hands with eight other people, because there are nine people total and they can't shake hands with themselves. The next person shakes hands with seven people, because they already shook hands with the first person. The third person shakes hands with six people, because they already shook hands with the first two. And so on and so forth. It ends up looking like this: 8+7+6+5+4+3+2+1= 36 This method can also be used to explain why the first method works. If you add up the first and last number, you get nine (8+1). The second and second to last number also equals nine (7+2), as does the third and third to last an fourth (6+3) and fourth to last (5+4). So the 9 x 8 part makes sense because you have eight numbers, and they all pair up to nine. You divide this number by two because their are two number in each pair that add up to nine. =========================================== I'll pick the first way and try to explain the reasoning: Partner-A in the handshake can be any one of 9 people. For each of those, Partner-B can be any one of the remaining 8 people. So there are (9 x 8) = 72 ways to have Partner-A shake hands with Partner-B. But they're not labelled and you can't tell 'A' from 'B'. (Roberts <---> Alito) is exactly the same as (Alito <---> Roberts). These two possibilities can only be counted as one. So the question is really asking for how many 'pairs' there are, regardless of who is 'A' and who is 'B'. If they were labelled with letters, there would be 72 different possibilities, but since the partners are equal, there are only half as many = 36 distinct pairs. In math-talk: There are 72 'permutations' ... possibilities where the order makes a difference ... and 36 'combinations' ... possibilities where the order doesn't matter.

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Q: If each of the 9 US Supreme Court justices shook hands with each of the other Justices once and only once how many handshakes would take place?
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