P&C

Asked by  | 11th Jan, 2010, 10:18: PM

Expert Answer:

There are 9 parallel horizontal lines and 9 parallel vertical lineson a chess board.

(1) To form a square: Two horizontal (a pair)and two vertical (a pair)  lines are  required which are lying at equal distance. This distance can be 1, 2,3,4......8 units.

Number of lines lying at unit distance = 8C1 = 8

Number of squares so formed of unit side = 8C1.8C1 = 82

Number of lines lying at two units distance = 7C = 7

Number of squares so formed of unit side = 7C1.7C1 = 72

Similarly number of square of three units , four units...................8 units will be 62, 52, 42, 32, 22 and  12 

So, total number of squares formed = 82 +  72 + 6+ 52 + 42 + 32 + 2+  12 

Using Σn2 = [(n(n+1)(2n+1))/6] , we get

82 +  72 + 6+ 52 + 42 + 32 + 2+  12 = 204

So, total number of squares formed on the chess board = 204

(2) To form a rectangle : Two horizontal (a pair)and two vertical (a pair)  lines are  required.

So, total number of rectangles formed = 9C2.9C2

                                                                     = 1296.

 

Answered by  | 12th Jan, 2010, 10:01: AM

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