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The number of rectangles that can be formed on a 8X8 chessbo
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Updated on: 09 Jul 2013, 07:04
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1. The number of rectangles that can be formed on a 8X8 chessboard isA. 2194 B. 1284 C. 1196 D. 1296 Answer: 2. The number of squares on 8X8 chessboard is A. 204 B. 220 C. 240 D. 210 Answer:
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Originally posted by saurabhgoel on 13 Sep 2010, 23:43.
Last edited by Bunuel on 09 Jul 2013, 07:04, edited 2 times in total.
Renamed the topic, edited the question and added the OAs.



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Re: Permutation & Combination
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Updated on: 14 Sep 2010, 05:25
For qtn2: # SQUARES answer = 1^2+2^2+3^2 . . . . 8^2 = 204 ...aplly the formula n(n+1)(2n+1)/6 for the SUM of squares of the first n #s ...n = 8 in this case).
Explanation:
To make a square from a 8by8 grid, we need to select equal # of grids from row and column.
8 adjucent grids from row and 8 adjucent grids from columns = 1*1 = 1^2 7 adjucent grids from row and 7 adjucent grids from columns = 2*2 = 2^2 . . . . 1 adjucent grid from row and 1 adjucent grid from columns = 8*8 = 8^2
total = 1^2+2^2+3^2 . . . . 8^2
Originally posted by muralimba on 14 Sep 2010, 05:13.
Last edited by muralimba on 14 Sep 2010, 05:25, edited 1 time in total.



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Re: Permutation & Combination
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14 Sep 2010, 05:24
No qtn 2 rectangles = (asuumed SUQARES(n by n) and RECTANGLES (m by n))
# of ways to select 1 grid from a cloumn = 8 now i can select 1/2/3/4/5/6/7/8 adjucent grids from the row to form a suqare/rectangle
hence # of ways = 8 (1+2+3+.....+8) so onnnn
total = 8 (1+2+3+.....+8) + 7 (1+2+3+.....+8) + ........ + 1 (1+2+3+.....+8)
= (1+2+3+.....+8) * (1+2+3+.....+8) = (1+2+3+.....+8)^2 = 36^2 = 1296
Hence answer = 1296 ((asuumed SUQARES(n by n) and RECTANGLES (m by n))
if asked for m by n (m not=n) rectangles only then answer = 1296204 (# of squares as calculated in the above post)
Hope it is clear



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Re: Permutation & Combination
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14 Sep 2010, 07:14



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Re: Permutation & Combination
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21 Sep 2010, 17:59
great explanation Bunuel, I would have never thought of it that way.



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Re: Permutation & Combination
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31 May 2014, 13:05
Unable to understand highlighted part: Bunuel wrote: Another way:
2. The number of squares on 8X8 chessboard is a) 204 b) 220 c) 240 d) 210
# of squares with are of 1*1=1 is 8*8=64;  this one is logical 8 rows 8 columns total 64, 1 unit area squares. # of squares with are of 2*2=4 is 7*7=49;  but 7x7 why and how to visualize this on board. # of squares with are of 3*3=9 is 6*6=36; ... # of squares with are of 8*8=64 is 1*1=1;
Total # of squares possible is 64+49+36+25+16+9+4+1=204.
Answer: A.
P.S. Not a GMAT questions
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Re: Permutation & Combination
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31 May 2014, 16:09
PiyushK wrote: Unable to understand highlighted part: Bunuel wrote: Another way:
2. The number of squares on 8X8 chessboard is a) 204 b) 220 c) 240 d) 210
# of squares with are of 1*1=1 is 8*8=64;  this one is logical 8 rows 8 columns total 64, 1 unit area squares. # of squares with are of 2*2=4 is 7*7=49;  but 7x7 why and how to visualize this on board. # of squares with are of 3*3=9 is 6*6=36; ... # of squares with are of 8*8=64 is 1*1=1;
Total # of squares possible is 64+49+36+25+16+9+4+1=204.
Answer: A.
P.S. Not a GMAT questions # of squares with are of 2*2=4 is 7*7=49; Rows: {1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 6}, {6, 7}, {7, 8} > 7 positions. Columns: {1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 6}, {6, 7}, {7, 8} > 7 positions. # of squares with are of 3*3=9 is 6*6=36; Rows: {1, 2, 3}, {2, 3, 4}, {3, 4, 5}, {4, 5, 6}, {5, 6, 7}, {6, 7, 8} > 6 positions. Columns: {1, 2, 3}, {2, 3, 4}, {3, 4, 5}, {4, 5, 6}, {5, 6, 7}, {6, 7, 8}> 6 positions. Not a GMAT question. So, you can ignore.
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Re: The number of rectangles that can be formed on a 8X8 chessbo
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03 Oct 2017, 14:13
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Re: The number of rectangles that can be formed on a 8X8 chessbo &nbs
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