Three are 12 points in a plane, no of three of which are in the same straight line, except 5 points whoich are collinear. Find
(i) the numbers of lines obtained from the pairs of these points.
(ii) the numbers of triangles that can be formed with vertices as these points.

Explanation

(i) No. of lines formed by joining the 12 points, taking 2 at a time `=.^(12)C_(2)`
No of lines formed by joining the 5 points taking 2 at a time `= .^(5)C_(2)`
But 5 collinear points when joined pairwise, give only one line
`:.` The required number of lines `= .^(12)C_(2) - .^(5)C_(2) +1`
`= 66 - 10 +1`
`= 57`
(ii) No of triangles formed by joining 12 points by taking 3 points at a time `= .^(12)C_(3)`
No. of triangles formed by joining 5 points by taking 3 points at a time `= .^(5)C_(3)`
But there is no triangle formed by joining 3 points out of 5 collinear points
`:.` The required number of triangles formed `=.^(12)C_(3)-.^(5)C_(3)`
`= 220 - 10 = 210`