InterviewSolution
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InterviewSolution
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In a plane there are 12 points, then answer the following questions. (a) Find the number of different straight lines that can be formed by joining these points, when no combination of 3 points are collinear. (b) Find the number of different straight lines that can be formed by joining these points, when 4 of these given points are collinear and no other combination of three points are collinear. (c) Find the number of different triangles that can be formed by joining these points, when no combination of 3 points are collinear. |
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Answer» (a) We know passing through two points in a plane we can draw only one line, i.e., we require to select any two points from the given 12 points which is possible in `.^(12)C_(2)` ways. `therefore` The number of different straight lines that can be formed by joining the given 12 points. `.^(12)C_(2)(12xx11)/(1xx2)=66`. (b) Given, out of the 12 points, 4 points are collinear. We know that collinear points form only one line. `therefore` These four points when they are not collinear will actually form `.^(4)C_(2)` lines, which are not forming here. `therefore` The number of the required lines `=.^(12)C_(2)-.^(4)C_(2)+1=66-6+1=61`. (c) We know, by joining three non-collinear points a triangle forms. `therefore` Three points can be selected from 12 points in `.^(12)C_(3)` ways. `therefore` The required number of triangles `=.^(12)C_(3)=(12xx11xx10)/(1xx2xx3)=220`. (d) Given 5 points are collinear. `rArr .^(5)C_(3)` triangles will not form. `therefore` The required number of triangles `=.^(12)C_(3)-.^(5)C_(3)=220-10=210`. (d) Given 5 points are collinear. `rArr .^(5)C_(3)` triangles will not form. `therefore` The required number of triangles `=.^(12)C_(3)-.^(5)C_(3)=220-10=210`. |
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