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  • If a group of 20 points, 12 of which are on the same line, can be used to select the vertices of triangles, how many triangles can you form?Option: 1</

If a group of 20 points, 12 of which are on the same line, can be used to select the vertices of triangles, how many triangles can you form?

Option: 1

1120

 


Option: 2

920


Option: 3

1040

 


Option: 4

1360 


Answers (1)

best_answer

Given that,

There are 20 points, out of which there are 12 points lying on the same line.

The triangle can be formed using 3 points.

The number of triangles formed from 20 points is given by,

\begin{aligned} &{ }^{20} C_3=\frac{20 !}{3 ! 17 !}\\ &{ }^{20} C_3=\frac{20 \times 19 \times 18}{3 \times 2}\\ &{ }^{20} C_3=20 \times 19 \times 3\\ &{ }^{20} C_3=1140 \end{aligned}

From the given 20 points, 12 are collinear.

So, from these 12 points, we cannot form any triangles.

Thus,

\begin{aligned} &{ }^{12} C_3=\frac{12 !}{3 ! 9 !}\\ &{ }^{12} C_3=\frac{12 \times 11 \times 10}{3 \times 2}\\ &{ }^{12} C_3=2 \times 11 \times 10\\ &{ }^{12} C_3=220 \end{aligned}

Hence, the required number triangle is,

\begin{aligned} &{ }^{20} C_3-{ }^{12} C_3=1140-220\\ &{ }^{20} C_3-{ }^{12} C_3=920 \end{aligned}

Therefore, the total number of triangles formed is 920.

 

 

Posted by

Pankaj

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