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Combinations — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsCombinations5 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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