Square Lattice Triangles
A 5 by 5 square lattice is formed by drilling holes in a piece of wood. Three pegs are placed in this lattice at random.
Find the probability that three randomly chosen points of a 5 by 5 lattice will form a triangle.
There are 25C3 = 2300 ways of picking three points from twenty-five. However, some of the sets will be collinear, and this can happen in a number of ways.
Along each vertical, horizontal, and main diagonal:
125C3 = 1210 = 120
Along the other diagonals:
Then, we have twelve more:
2300 (120+20+12) = 2148, hence the probability of three random points forming a triangle will be 2148/2300 = 537/575 0.934.
What about a 6 by 6 lattice?
Although there is no general formula, investigate other sized lattices to find an algorithm.