How many handshakes if 25 people shake hands with every other person in room?
Each person will shake hands with every other person, except
himself. If there are 25 people, each person will shake hands with
25-1 people, or 24. The number of times each person will shake
hands with another, will be 25x24. The number of handshakes will be
half of that, as each handshake is between two persons.
The formula, in other words, is x(x-1)/2, where x is the number
of people.
With 25 people, it will be 25x24/2 = 300 handshakes.