Everybody in a room shakes hands with everybo | vedclass

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Question

(B) Let the total number of persons in the room be $n$.Since every person shakes hands with every other person exactly once,the total number of handsh

Vedclass · Accepted Answer

$12$

Vedclass · Answer

(B) Let the total number of persons in the room be $n$.Since every person shakes hands with every other person exactly once,the total number of handshakes is given by the combination formula $^nC_2$,which represents choosing $2$ people out of $n$ to perform a handshake.Given that the total number of handshakes is $66$,we have the equation: $^nC_2 = 66$.Using the formula $^nC_2 = \frac{n(n-1)}{2}$,we get:$\frac{n(n-1)}{2} = 66$$n(n-1) = 132$$n^2 - n - 132 = 0$Factoring the quadratic equation: $(n - 12)(n + 11) = 0$.Since the number of persons $n$ must be positive,we have $n = 12$.Therefore,the total number of persons in the room is $12$.

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:

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