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a) 1/9

b) 2/7

c) 1/18

d) none
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1/18?
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Plz explain how 1/18.
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in chess board we have 8 rows and 8 columns.
take a row and see how many ways you can choose 2 adjacent boxes. you can choose it in 7 ways.
example if there are boxes form 1 to 8, number of  adj boxes =
(1,2)(2,3)(3,4)(4,5)(5,6)(6,7)(7,8) = 7
same happens with other 7 rows too. so from each row we can choose 2 adjacent boxes in 8*7 ways.
now, there can be boxes which share common edge beween the rows.
lets see how many adjacent boxes can be present in each column.
there will be 7 adj boxes in 1 column. so 8*7 adj boxes will be in all the columns.
so, total number of adj boxes= (8*7+8*7)C1
we are choosing 2 boxes from 64 boxes so 2*8*7/(64C2) = 1/18
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But it is also possible that there are 2 2×2 squares having a side in common. So we have to consider those cases also and proceed. Hence 1/18 should be wrong.

Similarly we have to check for 3×3 and 4×4 squares.

It is not possible to have 2 5×5 squares having a side in common since they will overlap if we try to do so.
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@Anusha,

If two squares are chosen at random on a chessboard the probability that they have at least one corner in common  ??

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i think i did not miss any 2 adjacent boxes. all the adj boxes in 2*2 ,3*3 and 4*4 squares are included in my calculation. you can check taking some small square like a 4*4 grid and compare ur answer and mine. both will be same :)
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@habib khan u are right 1/18 ans is only for 1x1 square..if we choose other we get none of these ans
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@motamarri anusha  u can refer the link if u have doubt.
http://www.careerbless.com/qna/discuss.php?questionid=2439

In each row we have 7 pairs of squares having a common side. So, totally 8*7 = 56 such squares horizontally. Similarly, we get 56 such squares vertically. So, total number of favorable cases = 56+56=112.
Required probability = 112/64C2
= 112*2/(64*63)

= 1/18

selected

In 64 squares,
4 at-corner squares, each has only 2 options to select from so    4*2C1
6*4 = 24 side squares, each has only 3 options to select from so    24*3C1
6*6 = 36 inner squares , each has 4 options to select from so    36*4C1

So.

(4 * 2 + 24 *3 + 36* 4)/(64*63) =  (224/4032) = 1/18

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