Everybody in a room shakes hands with everybody else. The total number of handshakes is 66. The total number of persons in the room are

Solution:

Let the total number of persons in a room be n since two persons make 1 handshake

∴ The number of handshakes = ⁿC₂

So ⁿC₂ = 66

⇒$\frac{{n!}}{{2!(n - 2)!}} = 66$

⇒$\frac{{n(n - 1)(n - 2)!}}{{2 \times 1 \times (n - 2)!}} = 66$

⇒$\frac{{n(n - 1)}}{2} = 66$

⇒ n² – n = 132

⇒ n² – n – 132 = 0

⇒ n² – 12n + 11n – 132 = 0

⇒ n(n – 12) + 11(n – 12) = 0

⇒ (n – 12)(n + 11) = 0

⇒ n – 12 = 0, n + 11 = 0

⇒ n = 12, n = – 11

∴ n = 12   .... (∵ n ≠ – 11)