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  • There are 1320 triangles formed by connecting 21 points in a plane where n points are in the same straight line. What is the numerical value of n?  <div class='qna-opti

There are 1320 triangles formed by connecting 21 points in a plane where n points are in the same straight line. What is the numerical value of n?

 

Option: 1

10


Option: 2

5


Option: 3

15


Option: 4

20


Answers (1)

best_answer

The number of triangles that can be formed is equal to the number of 3 non-collinear points that can be chosen.

The number of ways to choose three points from a possible 21 points is given by,

\mathrm{\begin{aligned} &{ }^{21} C_3=\frac{21 !}{3 ! 18 !}\\ &{ }^{21} C_3=\frac{21 \times 20 \times 19}{3 \times 2}\\ &{ }^{21} C_3=1330 \end{aligned}}

Assume that n points are collinear.

The number of ways to choose three points from a set of n collinear points is \mathrm{^{n}C_{3}}

?So, the number of ways to select 3 non-collinear points = The number of ways to select 3 points using all the points - The number of ways to select 3 points using the collinear points

Thus, the number of triangles formed is given by,

\mathrm{\begin{aligned} & N={ }^{21} C_3-{ }^n C_3 \\ & 1330=1320-{ }^n C_3 \\ & { }^n C_3=10 \end{aligned}}

Solving the above term to get,

\mathrm{\frac{n !}{3 !(n-3) !}=10 }

\mathrm{\frac{n(n-1)(n-2)}{6}=10 }

\mathrm{ n(n-1)(n-2)=60 }

\mathrm{ n=5 }

Therefore, the value of n is 5.

 




 

 

Posted by

Ritika Kankaria

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