In a party everyone shakes hand with each other. If total number of handshakes is 780 how many people were there?

There were 40 people at the party.

Let n be the number of people at the party.

Each person shakes hands with every other person, so each person shakes hands with (n - 1) people, a possible total of n(n - 1) handshakes.

But when person A shakes hands with person B, B also shakes hands with A, so each handshake would be counted twice.

→ number_of_handshakes = n(n - 1)/2

total number of handshakes is 780

→ n(n - 1)/2 = 780

→ n(n - 1) = 1560

→ n^2 - n - 1560 = 0

As 1560 is negative, one factor is positive and one is negative, so we need the factor pair of 1560 which has a difference of 1, namely: 39 x 40

→ (n - 40)(n + 39) = 0

→ n = 40 or -39

There cannot be a negative number of people → there are 40 people present.