Question
Two squats are chosen at random on a chessboard. The probability that they have a side in common is

1/9

2/7

1/18

None of these
easy
Solution
1/18
The number of ways of choosing the first square is 64, and that for the second square is 63. Therefore, the number of ways of choosing the first and second squares is 64 × 63 = 4032. Now we proceed to find the number of favorable ways. If the first square happens it be any of the four squares in the corner, the second square can be chosen in 2 ways. If the first square happens to be any of the 24 (non  corner) squares on either side of the chessboard, the second square can be chosen in 3 ways. If the first square happens to be any of the 36 remaining squares, the second square can be chosen in 4 ways. Therefore, the number of favorable ways is (4) (2) + (24) (3) + (36) (4) = 224. Thus, the probability of the required event is 224/4032 = 1/ 18.
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