probablity help?

Question

From a chess board three blocks are chosen at random. What is the probability that the three blocks are from same diagonal?

Answer 1

Total no. of ways of choosing three blocks out of 64 blocks, n = 64C3 = 41664.

Three blocks can be chosen from diagonals from left to right in 2 ( 3C3 + 4C3 + 5C3 + 6C3 + 7C3 ) + 8C3 ways = 196.

Three blocks can be chosen from diagonals from right to left in 196 ways.

Total number of ways of choosing three blocks such that they are from the same diagonal, r = 196 + 196 = 392.

By classical definition of probability, required probability = r / n = 392 / 41664 = 7 / 744.

Answer 2

Okay, given a diagonal of n squares, the number of ways 3 blocks can be chosen from it is n!/(n-3)!3!, or for diagonals of 3, 4, 5, 6, 7, 8 square lengths, we have 1, 4, 10, 20, 35, 56 ways, so that the total number of ways is 4(1+4+10+20+35) + 2(56) = 392 ways. But the number of ways 3 squares can be picked from a 8x8 chessboard is large, 64!/(61!3!) = 41,664. So there you have it, the odds of the three squares falling on a diagonal is 392/41664, or 7/744. A little less than 1%