If n lines are drawn in a plane such that no two of them are parallel and no three of them are concurrent, such that these lines divide the planein 67 parts, then find number of different points at which these lines will cut.

Explanation

Given number of straight lines=n, then
`1+underset(k=1)overset(n)(sum)k=67implies(n^(2)+n+2)/(2)=67`
`impliesn^(2)+n-132=0 implies(n+12)(n-11)=0`
`thereforen=11,n ne-12`
Hence, required number of points`=.^(n)C_(2)=.^(11)C_(2)=(11*10)/(2)`