If the squares of a $$8$$ x $$8$$ chessboard are painted either red or black at random. The probability that not all the squares in any column are alternating in color isThe probability that the chessboard contains equal number of red and black squares isThe probability that all the squares in any column are of same color and that of a row are of alternating cotror is

Question

If the squares of a $$8$$ x $$8$$ chessboard are painted either red or black at random. The probability that not all the squares in any column are alternating in color isThe probability that the chessboard contains equal number of red and black squares isThe probability that all the squares in any column are of same color and that of a row are of alternating cotror is

Infinity Learn · Accepted Answer

The total number of ways of painting first column when colors are not alternating is $$28$$ $$-$$ $$2$$. The total number of ways when no column has alternating colors is $$(28$$ $$-$$ $$2)8/264$$.The number of ways the square has equal number of red and black squares is  $$64C32$$. Hence, the required probability is  $$64C32/264$$.This is possible only when the column are alternating red and black. Hence, the required probability is $$2/264$$ $$=$$ $$1/263$$.

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