Three different squares are chosen randomly on a chessboard.
What is the probability that they lie in the same diagonal?
There are 64C3 ways of choosing 3 squares. This is the denominator of the probability.
The numerator of the probability is all of the possible ways of choosing 3 in the same diagonal. Obviously, the 1 and 2 length diagonals don't count. So we are left with: 4 3-length, 4 4-length, 4 5-length, 4 6-length, 4 7-length and 2 8-length diagonals. To find out all ways of choosing three squares in one particular diagonal, we can sum all the ways of choosing three squares in each diagonal.
Hence the numberator is: 4*(3C3) + 4*(4C3) + 4*(5C3) + 4*(6C3) + 4*(7C3) + 2*(8C3)
I don't have a calculator handy.