1)2^92)2^9-13)2^9-24)none of these

1.	1)2^92)2^9-13)2^9-24)none of these
Answer» Solution: For an m×n chessboard there are 2^m+2ⁿ−2 ways. Case I. There are two horizontally adjacent squares of the same color: 2^m−2 ways. Case II. There are two vertically adjacent squares of the same color: 2ⁿ−2 ways. Case III. None of the above: 2 ways. Hint for Case I: There are 2^m−2 ways to color one row so that two adjacent squares have the same color. The rest of the coloring is determined from that; colors must alternate in each column. (Note, therefore, that Cases I and II do not overlap.)

Answer»

Solution:

For an m×n chessboard there are 2^m+2ⁿ−2 ways.

Case I. There are two horizontally adjacent squares of the same color: 2^m−2 ways.

Case II. There are two vertically adjacent squares of the same color: 2ⁿ−2 ways.

Case III. None of the above: 2 ways.

Hint for Case I: There are 2^m−2 ways to color one row so that two adjacent squares have the same color.

The rest of the coloring is determined from that; colors must alternate in each column. (Note, therefore, that Cases I and II do not overlap.)

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