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lampat

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if 2 of the 64 square are choosen at random on the chess board , the probability that they have a side in common is
 
Hello, lampat!

If 2 of the 64 square are choosen at random on a chessboard,
the probability that they have a side in common is __.
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Choosing 2 of the 64 squares, there are:. \(\displaystyle {64\choose2} \,=\,2016\) possible outcomes.

Now we must count the number of "dominos" on the board.

The domino could be horizontal: .\(\displaystyle \square\!\square \)
There are 7 horizontal dominos in each row, and there are 8 rows.
There are: .\(\displaystyle 7\cdot8 \:=\:56\) horizontal dominos.

The domino could be vertical: .\(\displaystyle \begin{array}{c}\square \\ [-2mm] \square \end{array}\)
There are 7 vertical dominos in each column, and there are 8 columns.
There are: .\(\displaystyle 7\cdot8 \:=\:56\) vertical dominos.

Hence, there are: .\(\displaystyle 56 + 56 \:=\:112\) possible dominos.


Therefore: .\(\displaystyle P(\text{domino}) \:=\:\dfrac{112}{2016} \:=\:\dfrac{1}{18}\)
 
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