Question

Every two persons shake hands with each other in a party and the total number of handshakes is 66. The number of guests in a party is
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Answer 1

As far as handshakes go, for n guests there are (n - 1) + (n - 2) + (n - 3) + .... + 3 + 2 + 1 handshakes.

(Guest 1 shakes hands with the remaining n - 1 guests, Guest 2 shakes hands with the remaining n - 2 guests: We dont account for guest 1 here as she's already shaken hands with guest 1,.... and so on!)

This can be simplified to n x (n - 1) / 2

We know the total number of handshakes is 66

or n * (n - 1) / 2 = 66

or n * (n - 1) = 132

or n ^ 2 - n - 132 = 0

Getting the roots of this equation (of the form ax^2 + bx + c)

Roots = -b +- sqrt(b^2 - 4ac) / 2a

Here a = 1, b = -1 and c = -132

[Here sqrt = square root, +- is plus or minus, ^2 is square of ]

= (1 +- sqrt (1 - 4*132)) / 2 = (1 + sqrt (529))/2 and (1 - sqrt(529))/2

= -11 or 12.

-11 is not valid since we cannot have a negative number of people.

There fore there are 12 guests in the party