mental maths

${n \choose 2} = 66$

$n = 12$

**Solution: 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.

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 $\times$ (n-1))/2=66

so by solving the quadratic equation we get n=12

If there r 2 persons then 2*1/2=1 handshake,

for 3 persons 3*2/2=3 handshake,

for 4 persons 4*3/2=6 handshake,

for 5 persons 5*4/2=10 handshake,

& for n persons n*(n-1)=66 handshake (given),

that is 12*11=66,

hence 12 persons.