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# In how many ways can we choose two black squares on a chess board

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Director
Joined: 16 Jan 2019
Posts: 594
Location: India
Concentration: General Management
WE: Sales (Other)
In how many ways can we choose two black squares on a chess board  [#permalink]

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23 Feb 2020, 23:22
1
3
00:00

Difficulty:

95% (hard)

Question Stats:

21% (02:25) correct 79% (02:15) wrong based on 28 sessions

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In how many ways can we choose two black squares on a standard 8x8 chess board, so that they do not lie in the same row or column?

A. 240

B. 400

C. 496

D. 788

E. 1024
CrackVerbal Representative
Joined: 01 Mar 2019
Posts: 130
In how many ways can we choose two black squares on a chess board  [#permalink]

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Updated on: 22 Mar 2020, 04:01
Method 1:

Total number of black squares on the board = 32
Number of ways of selecting the first black square = $$32C_1$$ = 32
The selected black square will be common in the row and column from which we cannot select the second black square,
Therefore, the number of black squares unavailable for selecting the second = 4 in the row which cannot be used + 4 in the column which cannot be used - 1 common = 7
Therefore, number of black squares available for selecting the second = 32-7 = 25
Number of ways of selecting the second black square = $$25C_1$$ = 25

Therefore, total number of ways of selecting the 2 black squares = 32*25/(No. of repeating colours!) = 800/2! = 400 (we divide because the two squares are both black and hence the order of selection doesn't matter here)

Method 2:

Number of ways of selecting 2 black squares from the same row = $$4C_2*8$$ = 48
Number of ways of selecting 2 black squares from the same column = $$4C_2*8$$ = 48
Total number of incorrect ways of selecting 2 black squares (ie; such that they either lie in the same row or the same column) = 48 + 48 = 96
Total number of ways of selecting 2 black squares = $$32C_2$$ = $$\frac{32*31}{2*1}$$ = 496

Therefore, total number of ways of selecting the 2 black squares such that they don't lie in the same row or column = 496 - 96 = 400

Hope this helps.
_________________

Originally posted by svasan05 on 24 Feb 2020, 01:43.
Last edited by svasan05 on 22 Mar 2020, 04:01, edited 1 time in total.
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Joined: 24 Feb 2020
Posts: 1
Re: In how many ways can we choose two black squares on a chess board  [#permalink]

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21 Mar 2020, 22:25
svasan05

In method 1 there is a calculation mistake 32*25 <> 400. Thanks for method 2, a good approach
CrackVerbal Representative
Joined: 01 Mar 2019
Posts: 130
Re: In how many ways can we choose two black squares on a chess board  [#permalink]

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22 Mar 2020, 04:00
AnuragTiwari1991 wrote:
svasan05

In method 1 there is a calculation mistake 32*25 <> 400. Thanks for method 2, a good approach

Thanks Anurag, for pointing out.

I had missed out an important step. We must divide the final answer by 2 since the black squares are the same kind and hence the order of appearance doesn't matter (this is similar to forming words with repeating letters). This should lead to the correct answer.

I'll edit the solution to reflect this.
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Re: In how many ways can we choose two black squares on a chess board   [#permalink] 22 Mar 2020, 04:00
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