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Start out with small cases and you will see the pattern. At that point, you might even guess what the formula is. So let's ay the room has only A and B---two people. there is one handshake

(A with B). Three people (A, B, and C), there are three handshakes

(A with B and C; B with C). Now, let's try four people (A, B, C, and D), there are six handshakes

(A with B, C, and D; B with C and D; C with D). So if we have n+1 people, the number of handshakes is the sum of the first n numbers. This is n(n+1)/2 So for 6 people it is the sum of the first 5 numbers, which is 1+2+3+4+5=15 or 5x6/2=30/2=15 Another way to think of this, if you have done some combinatorics, is for 6 people, in how many ways can you pick two. This is 6C2 or 6x5/2= 15

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