∵ Straight line can be formed by joining two points & triangle can be formed by joining three non-collinear points.
(i) Number of ways in which 2 points are selected from 13 points = \(^{13}C_2\).
But given that 6 points are collinear (in straight line).
Therefore these 6 points can form only one line.
So, we have to subtract (\(^6C_2\) - 1) from the ways of selection 2 points from 13 points.
∴ Total straight lines that can be possible
= \(^{13}C_2\) - (\(^6C_2\) - 1)
= \(\frac{13\times 12}{2}\) - \(\frac{6\times 5}{2}\) + 1
= 13 x 6 - 15 + 1
= 78 - 14
= 64.
(ii) Number of ways in which 3 non-collinear points are selected from 13 points = \(^{13}C_3\).
But given that 6 points are collinear that never be formed any triangle.
So, we have to subtract the selection of 3 non-collinear points from these 6 points.
So, total number of triangle = \(^{13}C_3\) - \(^{6}C_3\)
= \(\frac{13\times 12\times 11}{3\times 2}\) - \(\frac{6\times 5\times 4}{3\times 2}\)
= 286 - 20
= 266.
