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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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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

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Total possible triangles = $\binom{16}{3}$ = 560

Now we know that if all points are collinear then the triangle is not possible.

So we will simply deduct the cases wherein we are selecting 3 points which are collinear,

How many groups of 4 collinear points do we have?

4 horizontal lines + 4 vertical lines + 2 diagonals = 10

So $10*\binom{4}{3}=40$ invalid counts

How many groups of 3 collinear points do we have?

Only 4.

So $4*\binom{3}{3}=4$ invalid counts

Therefore the total number of triangles =

560 – (40+4) = 516

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