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Question
There are 15 points in a plane , no three of which are in the same straight line with exception of 6, which are in the same straight line. Find the number of .a) straight lines formed by) number of triangles formed by joining these points.
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Solution
(i). If no three points out of 15 points lie on a line, then the number of lines is 15C2 as exactly one line is drawn through two points. Since 6 out of 15 points are collinear, they from only one line instead of 6C2 lines. Hence the number of lines gets decreased by 6C2 -1. Hence the required number of lines
= 15C2 -(6C2 -1) = 105 -15+1 = 91
(ii). A triangle is formed by joining 3 non-collinear points. So if the 15 points are all non-collinear, 15C3 triangles can be formed. But here 4 points lie on a line and form no triangle instead of 6C3. Thus the required number of triangles
= 15C3 -6C3 = 455 -20 = 435
(i). If no three points out of 15 points lie on a line, then the number of lines is 15C2 as exactly one line is drawn through two points. Since 6 out of 15 points are collinear, they from only one line instead of 6C2 lines. Hence the number of lines gets decreased by 6C2 -1. Hence the required number of lines
= 15C2 -(6C2 -1) = 105 -15+1 = 91
(ii). A triangle is formed by joining 3 non-collinear points. So if the 15 points are all non-collinear, 15C3 triangles can be formed. But here 4 points lie on a line and form no triangle instead of 6C3. Thus the required number of triangles
= 15C3 -6C3 = 455 -20 = 435
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