Every body in a room shakes hands with every body else. The total number of hand shakes is $$66$$. The total number of persons in the room is

Question

Infinity Learn · Accepted Answer

Let total number of persons be n.Since two persons shake hands, so number of ways of shaking hands $$=nC2$$.According to question, we have  $$nC2=66or$$ n!(n$$−$$2)!$$2$$!$$=66$$  or  $$n(n$$−$$1)(n$$−$$2)$$!(n$$−$$2)!$$2$$×$$1=66or$$ $$n2$$−$$n=132$$ or $$n2$$−n−$$132=0or$$ $$n2$$−$$12n+11n$$−$$132=0or$$ $$n(n$$−$$12)+11(n$$−$$12)=0$$⇒$$n=$$−$$11$$ or $$12But$$, n $$=$$ –$$11$$ is not possible.∴ $$n=12$$.

Every body in a room shakes hands with every body else. The total number of hand shakes is 66. The total number of persons in the room is

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