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Combinations — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSCombinations4 Marks
Question
Out of 18 points in a plane, no three are in the same straight line except five points which are collinear. How many:
  1. Straight lines.
  2. Triangles can be formed by joining them?
✓

Answer

There are 8 points in a plane out of which 5 points are collinear. Then number of strianght lines joining these points are, $\Rightarrow {^\text{n}}\text{C}_{\text{2}}-({^\text{p}}\text{C}_{\text{2}}-1)$ $\Rightarrow {^\text{n}}\text{C}_{\text{2}}-{^\text{p}}\text{C}_{\text{2}}+1$ $\Rightarrow{^\text{18}}\text{C}_{\text{2}}-{^\text{5}}\text{C}_{\text{2}}+1$ $\Rightarrow \frac{18\times17}{2}-\frac{5\times4}{2}+1$ $\Rightarrow144$ Number of triangle $={^\text{13}}\text{C}_{\text{3}}$ $=\frac{13!}{3!10!}=\frac{13\times12\times11}{3\times2}$ $=13\times2\times11$ $=13\times22$ $=806$

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