At a party, everyone shook hands with everybody else. There are 66 handshakes. How many people were there in the party?
According to the rule of combination we know nCr = n! / (n-r)! r!
so, n(n-1)(n-2)! / (n-2)! 2! = 66 [study combination online if don't know how it works]
=> n(n-1) = 66 x 2 = 132
=> n² - n - 132 = 0
=> n² - 12n + 11n - 132 = 0 [there are thumb rule how to middle term break any big integer]
=> (n-12)(n+11) = 0
=> n = 12, -11
n cannot be negative so ans is 12.
According to the rule of combination we know nCr = n! / (n-r)! r!
so, n(n-1)(n-2)! / (n-2)! 2! = 66 [study combination online if don't know how it works]
=> n(n-1) = 66 x 2 = 132
=> n² - n - 132 = 0
=> n² - 12n + 11n - 132 = 0 [there are thumb rule how to middle term break any big integer]
=> (n-12)(n+11) = 0
=> n = 12, -11
n cannot be negative so ans is 12.
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