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in the chessboard, There are 9 horizontal lines and 9 vertical lines. Choose two distinct horizontal lines, and two distinct vertical lines. (any rectangle determines a pair of horizontal lines and a pair of vertical lines)

so # rectangle = ^{9}C_{2} * ^{9}C_{2 }= 1296

u can see that there are 8^{2} small 1×1 squares, 7^{2} 2×2 squares, and so on down to 1^{2 }1×1 squares,,, u need to add to find total # squres

so # squres = 1^{2 }+ 2^{2} + 3^{2} + 4^{2 }+ 5^{2 }+ 6^{2 }+ 7^{2 }+ 8^{2 }= 204

number of rectangles in a chess board that are not sqaure = 1296 - 204 = 1092

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Explanation:-

No of Squares :-

Dimension | No of Squares |

1X1 | 64 |

2X2 | 49 |

3X3 | 36 |

4X4 | 25 |

5X5 | 16 |

6X6 | 9 |

7X7 | 4 |

8X8 | 1 |

So no of squares = 204

No of Rectangles :-

In chess board we have 9 vertical lines and 9 horizontal line, now to construct rectangle of any dimension we have to take 2 horizontal line and 2 vetical line, which can be done in

= C(9,2) * C(9,2)

= 36 * 36

= 1296

To get total no of Rectangles which are not square simply subtract total no of squares from total no of rectangle.

= **Total no of rectangles - total no of squares**

= 1296 - 204

= 1092

So, answer is **1092**

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