How Many Triangles Are In A 5 by 5 Square?

Answer 1

If you have a grid of 5x5 squares you will have no triangles at all since there are no diagonal lines.

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Here is a more likely answer - though I am not sure it is totally correct.

A triangle is defined by three vertices. The total number of ways of picking 3 out of 25 points is 25*24*23/(3*2*1)

25 ways of picking a point from the grid at random, 24 ways to pick another point from the grid and 23 to pick the third. Divide by 3*2*1 since the order in which these collinear points is picked does not matter.

However, some of these are not triangles: they are not triangles if all three points are in the same row or in the same column.

Number of ways for that = 2*(25*4*3)/(3*2*1)

25 ways of picking a point from the grid at random, 4 ways of another point in the same row, 3 ways of picking the third in the same row. Divide by 3*2*1 since the order in which these collinear points is picked does not matter. Multiply by two because the same argument applies for collinearity in columns.

So answer = 2300 - 100 = 2200