There are n straight lines in a plane such that `n_(1)` of them are parallel in onne direction, `n_(2)` are parallel in different direction and so on, `n_(k)` are parallel in another direction such that `n_(1)+n_(2)+ . .+n_(k)=n`. Also, no three of the given lines meet at a point. prove that the total number of points of intersection is `(1)/(2){n^(2)-sum_(r=1)^(k)n_(r)^(2)}`.

Explanation

Total number of points of intersection when no two of n gien lines are parallel and no three of them are concurrent, is `.^(n)C_(2)`. But it is given that there are k sets of
`n_(1),n_(2),n_(3), . .,_n_(k)` parallel lines such that no line in one set is parallel to line in another set.
Hence, total number of points of intersection
`=.^(n)C_(2)-(.^(n_(1))C_(2)+.^(n_(2))C_(2)+ . . +.^(n_(k))C_(2))`
`=(n(n-1))/(2)-{(n_(1)(n_(1)-1))/(2)+(n_(2)(n_(2)-1))/(2)+ . . .+(n_(k)(n_(k)-1))/(2)}`
`=(n(n-1))/(2)-(1)/(2){(n_(1)^(2)+n_(2)^(2)+ . .+n_(k)^(2))-(n_(1)+n_(2)+ . . .+n_(k))}`
`=(n(n-1))/(2)-(1)/(2){underset(r=1)overset(k)(sum)n_(r)^(2)-n}`
`=(n^(2))/(2)-(1)/(2) underset(r=1)overset(k)(sum)n_(r)^(2)=(1)/(2){n^(2)-underset(r=1)overset(k)(sum)n_(r)^(2)}`.