Hand-Shake Problem !!

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 n^2+ 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. The last number that you entered (11) is n.

(n-1)+(n-2)+-------------------------------+3+2+1=66

$\Rightarrow$ (n-1)*n/2=66

$\Rightarrow$ n=12,-11

Ans: 12

n=12

Another way to solve it: C(n,r) = 66, with r=2. Using combinatorics, C(n,2) means we take from n elements, 2 by 2, representing every handshake (or a pair of contacts). Looking at Pascal´s Triangle, we see that 66 is on the 12th line and 2nd column (where n represent´s the lines and r=2 is the triangle´s columns).

n(n-1)/2 =66. Thus by solving this we get n=12