In a party every person shakes hands with every other person. If there are 55 hands shakes, find the number of person in the party.

Question: In a party every person shakes hands with every other person. If there are 55 hands shakes, find the number of person in the party.

Solution:
Let n be the number of persons in the party
Total number of hands shake is given by ⁿC₂

so,
ⁿC₂ = 55
⇒ n!/2! (n - 2)! = 55
⇒ n(n - 1)/2 = 55
⇒ n2 - n = 110
⇒ n2 - n - 110 = 0
⇒ n2 - 11n + 10n - 110 = 0
⇒ n(n - 11) + 10(n - 11) = 0
⇒ (n - 11) (n + 10) = 0

∴ n + 10 = 0
n = - 10, not possible
∴ n - 11 = 0
n = 11

so, there are 11 persons.