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Question
Everybody in a room shakes hands with everybody else. The total number of handshakes is 66. The total number of persons in the room is ______.
Options
11
12
13
14
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Solution
Everybody in a room shakes hands with everybody else. The total number of handshakes is 66. The total number of persons in the room is 12.
Explanation:
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
⇒ `(n!)/(2!(n - 2)!)` = 66
⇒ `(n(n - 1)(n - 2)!)/(2 xx 1 xx (n - 2)1)` = 66
⇒ `(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)
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