At the party

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.

Jaber Al-arbash by Jaber Al-arbash , 28, USA

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2 solutions

Jaber Al-arbash
Oct 2, 2014

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.

Since 66 is a relatively small number, you can also solve this problem with a hand calculator. Add 1 + 2 = + 3 = +... etc. until the total is 66

3 Helpful 0 Interesting 0 Brilliant 0 Confused

I made a table with the number of guests corresponding with the number of handshakes. I also figured out the general formula too, but I didn't know how to write out a formal way of solving it using the general formula.

William Li - 6 years, 8 months ago

be the persons: A, B, C, D... The number of handshakes will be AB, AC, AD,... or simply C(x,2)

Claudio Felipe - 6 years, 7 months ago
Rahul Chauhan
Jan 6, 2015

For smaller no. as 66 the method suggested by Jaber Al- arbash is good but in general Permutation and combinations is what you need. let n people meet in a room and each shake hand with all others. there will be as many hand shakes as there are as many combination of n different things taken 2 at a time. {Here order is not important}. . Mathematically nC2 =66 = n (n-1) / 2 = 66 Solving further we get n=12

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