At a reunion everyone shakes everyone else's hand once. Not one person shakes someone else's hand twice. If there were N hand shakes, how many people were at the reunion.
The problem is being solved for a total of N hand shakes. This can be used to determine the number of people at the reunion for any value of N.
Let the total number of people equal X. Any person that is selected shakes hands with everyone else. This gives the number of people that the person shakes hands with as (X - 1). This is applicable for each of the X people that attend the reunion. But as each hand shake involves the participation of 2 people, the total number of hand shakes is reduced by a factor of 2. The total number of hand shakes if there are X people at the reunion is (X*(X - 1))/2. Equating this to N and solving for X gives the number of people at the reunion if there were a total of N hand shakes.
The number of people X at the reunion if the total number of hand shakes is N can be found by solving (X*(X - 1))/2 = N