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  • How many triangles can be formed if a collection of 35 points, 17 of which are on the same line, can be used to choose the triangle vertices?  <

How many triangles can be formed if a collection of 35 points, 17 of which are on the same line, can be used to choose the triangle vertices?

 

Option: 1

3765


Option: 2

4825


Option: 3

5865


Option: 4

6125


Answers (1)

best_answer

Given that,

There are 35 points total, 17 of which are located along the same line.

Three points can be used to create the triangle.

The 35 points form a triangle, and its number is given by,

\mathrm{\begin{aligned} & { }^{35} C_3=\frac{35 !}{3 ! 32 !} \\ & { }^{35} C_3=\frac{35 \times 34 \times 33}{3 \times 2} \\ & { }^{35} C_3=35 \times 17 \times 11 \\ & { }^{35} C_3=6545 \end{aligned}}

From the given 35 points, 17 are collinear.

So, from these 17 points, we cannot form any triangles.

Thus,

\mathrm{\begin{aligned} &{ }^{17} C_3=\frac{17 !}{3 ! 14 !}\\ &{ }^{17} C_3=\frac{17 \times 16 \times 15}{3 \times 2}\\ &{ }^{17} C_3=17 \times 8 \times 5\\ &{ }^{17} C_3=680 \end{aligned}}

Hence, the required number triangle is,

\mathrm{\begin{aligned} & { }^{35} C_3-{ }^{17} C_3=6545-680 \\ & { }^{35} C_3-{ }^{17} C_3=5865 \end{aligned}}

Therefore, the total number of triangles formed is 5865.

 

 

Posted by

himanshu.meshram

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