Question: Three 1x1 squares are taken on a 8x8 chess board.

1. What is the probablity, that all three squares are diagonal to each other ?

2.What is the probability, that all three squares are diagonal to each other and lie adjacent.


Solution:

Answer 1.
Total ways of placing 3 squares on the 8 x 8 grid is 64c3
Total ways in which all 3 squares are diagonal to each other are:
1st and 2nd diagonal rejected becoz in this there are less than 3 squares.
In the remaining diagonals the possible ways are 3c3, 4c3, 5c3, 6c3, 7c3, 8c3
All the ways are repeated again except 8c3.
Similar set of ways are their when we start from lower left corner.
So total number of ways are 2*( 2*(3c3+4c3+5c3+6c3+7c3) + 8c3) = 392
Hence the probability = 392/64c3.

Answer 2.
Total ways of placing squares such that they are adjacent and diagonal to each other are:
From the diagonal with 3 places = 1
......................... with 4 places = 2
.............................. with 5 places = 3
Hence total ways are 2*( 2*( 1 + 2 + 3 + 4 +5) + 6 ) = 72
Hence the probability would be = 72/64c3