If you have 1013 people and all must shake hands with everyone how many handshakes occur?

Question

They only shake hands with once

Answer 1

1,026,169 handshakes

Answer 2

Let's start by looking at a smaller number: Take 3 people. Each person shakes 2 other hands, so there are 3*2 = 6 counted shakes. But in actuality, only 3 shakes occured. If you have 5 people, there are 5*4 = 20 counted shakes, but only 10 occured. Using mathematical induction, we see that the number of counted shakes equals n * n-1, where n is the number of people. Thus, for the example above, 1013 * 1012 = 1025156 counted shakes, and therefore 512578 shakes occured.

This question falls under an area of math called Graph Theory. Essentially, the problem can be illustrated by drawing a circle of 1013 dots. Connect each dot to every other dot. That illustrates the handshakes. The connecting lines are called "edges" and the dots are called "vertices." The collection of edges and vertices is called a "graph." The number of lines that touch each vertice is called the "degree" of the vertice. In this problem, each vertice has a degree of 1012. There is a theorem in graph theory that states that the sum of the degrees of a graph is equal to twice the number of edges. So, adding 1012 up 1013 times (1012 * 1013), you get 1025156 as the sum of the degrees. Therefore, there are half that many edges, or 512578 edges. Therefore, there were 512578 shakes.

Answer 3

1013 x 1012 = 1,025,156

you don't shake hands with yourself.

or

1013 x 1013 = 1,026,169 - 1 = 1,026,168.

hrmm, maybe i should just quit .. LOL

Answer 4

1012+1011+1010+1009...

Basically it's the sum of a 1,012 digit arithmetic series in which the common difference is -1 and the first number is 1,012

It starts at 1,012 because you don't shake with yourself.

This works out to exactly 512,578 handshakes.

Answer 5

1013 x 1013 = 1,026,169

Answer 6

1013 factorial which is a very large number

Answer 7

One big shake, if everyone's got their hands together in a giant circle.
The mechanics of it are a bit overwhelming but it could be done.

Answer 8

1, 026,169 handshakes.

Answer 9

513,591 hand shakes.

Answer 10

what he said :)