# The Handshake Problem

I have seen in facebook groups that some people are not really familiar how to solve handshake problems.  I have prepared visuals to make sure that after reading this article it will be easy for you to solve this type of problem. Assuming there are 4 people namely A,B,C, and D. Each has to have a handshake with each other once. How many handshakes are there?

“A” can have a handshake to B,C,D in three ways. “B” can also have handshakes in 3 ways. “C” can also have handshakes in 3 ways. “D” can also have handshakes in 3 ways. If we count the number of handshakes happened there are 12, However  if we list down all handshakes happened we counted each handshakes twice. So we need to divide 12 by 2 in order to get the correct number of handshakes which is 6.

Worked Problem 1:

There are 10 contestants joined the National Math Challenge finals. Each of them had a handshake with each other once. How many handshakes happened?

Solution:

Each person made 9 handshakes with each other. Since there are 10 persons, there are 10(9) or 90 handshakes happened but again, we counted each handshakes twice so we divide it by 2. The actual handshakes happened is only 45.

Using combinations, this problem can be easily solved by the following formula; $H = \displaystyle\frac{n(n-1)}{2}$ or nC2

Where n is the number of persons and is the number of handshakes.

Worked Problem 2:

In an acquaintance party of witches and wizards, each witches and wizards needs to handshake with each other. If there are 105 handshakes happened. How many people are there in the party?

Using the formula given, $H=nC2$ $105=\displaystyle\frac{n(n-1)}{2}$ $210 = n^2-n$ $n^2-n-210=0$ $(n-15)(n+14)$

n=15 is the only solution. Therefore, there are 15 people in the party.