Hand Shakes!

Probability Level 2

At a party, everyone shook hands with everybody else. There were 66 handshakes. How many people were at the party?


The answer is 12.

Aniket Paul by Aniket Paul , 22, India

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2 solutions

Call the number of people is x x

There were 66 handshakes, it means there were [ x ( x + 1 ) ] 2 = 66 \displaystyle \frac{[x(x+1)]}{2}=66

x ( x + 1 ) = 132 x(x+1)=132

x 2 + x = 132 x^2 + x=132

Therefore, the number best fits in x x is 12 \boxed{12}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

It doesnt fit. its a minus instead of a sum for it to fit

Esteban Velàsquez - 5 years, 9 months ago
Aniket Paul
Jun 8, 2015

12

In general, with n+1 people, the number of handshakes is the sum of the first n consecutive numbers: 1+2+3+ ... + n. Since this sum is n(n+1)/2, we need to solve the equation n(n+1)/2 = 66. This is the quadratic equation n2+ n -132 = 0. Solving for n, we obtain 11 as the answer and deduce that there were 12 people at the party.

Add 1 + 2 = + 3 = +... etc. until the total is 66. The last number that you entered (11) is n.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

using a calculator would be foolish for such a small sequence I think.

Nelson Mandela - 6 years ago

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i know that.....

Aniket Paul - 6 years ago

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