In a party, every person shakes hands with every other person. If there occur a total of 120 handshakes in the party, how many people were present at the party?

In a party, every person shakes hands with every other person. If there occur a total of 120 handshakes in the party, how many people were present at the party? Correct Answer 16

Let n be the number of people in the party.

Number of handshakes = 120

Total number of handshakes is given by ⁿC₂ ⁿC₂ = 120

⇒ n!/ = 120

⇒ n × (n - 1)/2 = 120

⇒ n² - n = 240

⇒ n²- n - 240 = 0

⇒ n² - 16n + 15n – 240 = 0

⇒ n(n – 16) + 15(n – 16) = 0

⇒ (n – 16) (n + 15) = 0

Then, n = 16 or n = -15.

We will ignore negative value of n as number of people cannot be n.

∴ Number of people = 16