If 10 people shake hands with each other, how many total handshakes will there be?

Hello, you are solving formulas in this case, heres a good way of doing these problems: :)

We know that that 10 people shake hands, so each person shakes 9 hands. The first person shakes 9 hands, the second person shakes 8 hands, the third person shakes 7 hands, and so on. That would give you the answer, however in the case of bigger numbers, this technique is not the best, in fact the best is by doing the following. 

Each handshake consists of two people shaking hands. How many ways can you choose those two people if there are 10 people in the group? You can choose any one of 10 people for the first person involved in the handshake; then for each of those 10 choices, you have 9 choices for the second person. So you have 10*9 = 90 ways of choosing the two people for the handshake. But wait a minute. Person A shaking hands with person B is the same as person B shaking hands with person A; this means that in counting the  90 handshakes you have actually counted each one twice, so the actual number of handshakes is half of 90, or 45.

I hope i helped, and your final answer is 45

Thank you,
Darian