Out of 18 points in a plane,no three are in t | vedclass

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(C) Total points $n = 18$. Number of collinear points $m = 5$.$(i)$ Number of straight lines: The number of lines formed by $n$ points is $^{n}C_{2}$.

Vedclass · Accepted Answer

$(i) 144, (ii) 806$

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(C) Total points $n = 18$. Number of collinear points $m = 5$.$(i)$ Number of straight lines: The number of lines formed by $n$ points is $^{n}C_{2}$. Since $m$ points are collinear,they form only $1$ line instead of $^{m}C_{2}$ lines. Thus,the formula is $^{n}C_{2} - ^{m}C_{2} + 1$.Calculation: $^{18}C_{2} - ^{5}C_{2} + 1 = \frac{18 	imes 17}{2} - \frac{5 	imes 4}{2} + 1 = 153 - 10 + 1 = 144$.$(ii)$ Number of triangles: The number of triangles formed by $n$ points is $^{n}C_{3}$. Since $m$ points are collinear,they do not form any triangle. Thus,the formula is $^{n}C_{3} - ^{m}C_{3}$.Calculation: $^{18}C_{3} - ^{5}C_{3} = \frac{18 	imes 17 	imes 16}{3 	imes 2 	imes 1} - \frac{5 	imes 4 	imes 3}{3 	imes 2 	imes 1} = 816 - 10 = 806$.

Out of $18$ points in a plane,no three are in the same straight line except five points which are collinear. The number of $(i)$ straight lines,$(ii)$ triangles which can be formed by joining them is

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