# P and C doubt

1 vote
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Given a 4 *4 grid  points , how many Triangles with vertices on the grid can be formed?
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144  ??
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1 vote

I have calculated it as 516.

The selection of 3 points out of 16 can be done in 16C3 = 560.

This include points that are collinear. If you calculate the number of collinear lines with 3 grid points then you will get 44 such lines.

So, final number of possible triangles are 560 -44 = 516

This stackexchange Q&A helped me to solve this: https://math.stackexchange.com/a/634131

selected
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16C3 is total 3 points selected. -560

10* 4C3 - collinear lines 4 vertical 4 horizontal 2 diagonal -40

I m unable to visualise the other 4 collinear lines

Can u plzz help me out

I think I m missing something...
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Please consider counting these 4 lines too  0
Thanku....:)
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For 5* 5 grid

Will it be like this

25C3  all 3 points selected

12* 5C3 = 120   collinear lines 5 each horizontal vertical and 2 diagonal

4 non diagonal collinear lines with 3  grid points

4* 4C3 non diagonal collinear lines with 4 points

I m getting 2160...

But there how they got 12 collinear lines with 3 grid points....:(

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