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Permutation and Combinations — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsPermutation and Combinations3 Marks
Question
Out of $18$ points in a plane, no three are in the same line except five points which are collinear. Find the number of lines that can be formed joining the point.
$[$Hint: Number of straight lines $= ^{18}C_2 – ^5C_2+ 1.]$
✓

Answer

Total number of points $= 18$ Out of $18$ number$, 5$ are collinear and we get a straight line by joining any two points.
$\therefore$ Total number of straight line formed by joining $2$ points out of $18$ points $= ^{18}C_2$
Number of straight lines formed by joining $2$ points out of $5$ points $= ^5C_2$
But $5$ points are collinear and we get only one line when they are joined pair wise.
So$,$ the required number of straight lines are
$=\ ^{18}\text{C}_2-\ ^55_2+1=\frac{18.17}{2.1}-\frac{5.4}{2.1}+1=153-10+1=144$
Hence$,$ the total number of straight line $= 144$

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