At a business meeting every person shakes each other's hand once if they were 91 handshakes in total the number of people at the meet is

This is basically asking you to find the number of vertices of a complete graph that has 91 edges. (I've attached an example of one where , taken from Wiki "complete graph" article)

Let's say you have  unconnected vertices, each labelled with a number (just for the purpose of tracking which ones you've already taken into account). Draw edges connecting  to all the other vertices. There are possible edges you can draw.

Now starting from , there's already an edge between it and , which means there are other possible edges that you can draw.

Next, from , you can only draw new edges because is already connected to and .

Continuing in this pattern, you find that the second-to-last edge, can only be connected once to , and finally arriving at you find that all possible connections have been exhausted.

The takeaway here is that the total number of possible connections is the sum



Writing the terms in reverse order, you have



So now if you add up the first, second, ..., terms of the two equivalent sums, you have




In other words, you're adding up , times (since each corresponds to a distinct vertex), so you have



Dividing by 2, you end up with



Now, since you know that 91 handshakes took place in total, you have



Obviously, we omit the negative solution, so there were 13 people at the meeting.