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parveen110: Senior | Next Rank: 100 Posts; Posts: 91; Joined: Fri Jan 17, 2014 7:34 am; Thanked: 7 times

by parveen110 » Tue Jun 24, 2014 4:38 am

In how many ways is it possible to choose two black squares on a 8 Ã— 8 chessboard so that the squares do not lie in the same row or same column?

a. 400
b. 200
c. 800
d. 100
e. 475

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Source: Beat The GMAT — Problem Solving | Tue Jun 24, 2014 4:38 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Tue Jun 24, 2014 5:52 am

parveen110 wrote:In how many ways is it possible to choose two black squares on a 8 Ã— 8 chessboard so that the squares do not lie in the same row or same column?

a. 400
b. 200
c. 800
d. 100
e. 475

The GMAT would not expect a test-taker to know the structure of a chessboard, which looks as follows:

First black square:
The total number of squares = 8*8 = 64.
Half are black, half are white.
Number of options for the first black square = 64/2 = 32.

Second black square:

The figure above illustrates the number of options for the second black square.
If the first black square selected is the square highlighted in blue, then none of the squares highlighted in red may be chosen as the second black square.
Remaining options for the second black square = 32 - (1 blue square) - (6 red squares) = 25.

To combine the options above, we multiply:
32*25.
Since the order of the two squares doesn't matter, we divide by the number of ways the two squares can be ARRANGED (2!):
(32*25)/(2*1) = 400.

The correct answer is A.

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pmFollowed by: 24 members; Location: New Delhi, India; Thanked: 205 times

by GMATinsight » Tue Jun 24, 2014 9:38 am

An Alternate and Easy approach

Two square blocks from one row can be selected in 4C2 ways because there are 4 black squares in every row and we are taking the total ways to choose two squares from those 4 squares
4C2 = 6

Since there are 8 rows so total ways of choosing two squares from any one row out of 8 rows = 6 x 8 = 48

Similarly Total ways of choosing two squares from any one row out of 8 rows = 6 x 8 = 48
[Because Number of rows = Number of column, therefore ways of choosing 2 square will be same even in columns as were in rows]

Total UNFAVORABLE ways = 48 + 48 = 96

Total Possible outcomes of choosing two Black squares in a chess board = 32C2 = 496

Total Favorable ways of choosing 2 Black squares so that they don't lie in same row or same column = 496-96 = 400 ANSWER

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by Brent@GMATPrepNow » Tue Jun 24, 2014 12:30 pm

parveen110 wrote:In how many ways is it possible to choose two black squares on a 8 Ã— 8 chessboard so that the squares do not lie in the same row or same column?

a. 400
b. 200
c. 800
d. 100
e. 475

I solved this question using the same technique as Mitch's.
However, here's another approach that works:

First IGNORE the rule that says the squares must not lie in the same row or same column.
So, we need only count the number of ways that we can select 2 black squares from 32 black squares.
Since the order of the selected squares doesn't matter, we can use combinations.
We can select 2 squares from 32 squares in 32C2 ways
32C2 = (32)(31)/2 = (16)(31)

IMPORTANT: we need not evaluate (16)(31). You'll see why shortly.

Aside: If anyone is interested, we have a free video on calculating combinations (like 32C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

Since we have IGNORED the rule about squares not lying in the same row or same column, our total of (16)(31) possibilities includes some scenarios that BREAK the rule and some scenarios that FOLLOW the rule. We're going to determine the PROPORTION of scenarios that FOLLOW the rule.

PROPORTION of scenarios that FOLLOW the rule
I'll borrow Mitch's diagram for this.

Once we select 1 black square (like the square highlighted in blue above), there are 31 black squares remaining. Of those 31 black squares, 6 BREAK the rule and 25 FOLLOW the rule (about squares not lying in the same row or same column).
In other words, for each of the (16)(31) possibilities that we calculated above, 25/31 of them FOLLOW the rule.

So, the # of possibilities that FOLLOW the rule = (16)(31)(25/31) = (16)(25) = [spoiler]400 = A[/spoiler]

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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[email protected]: Elite Legendary Member; Posts: 10392; Joined: Sun Jun 23, 2013 6:38 pmFollowed by: 511 members; Location: Palo Alto, CA; Thanked: 2867 times; GMAT Score:800

by [email protected] » Tue Jun 24, 2014 6:53 pm

HI parveen110,

Under normal circumstances, I would approach this question in much the same way that Mitch approached it.

However, this question has an interesting "design shortcut" to it that would help you to answer it quickly if you were low on time.

Since the chessboard has half white squares and half black squares, we know that there are 32 of each. When selecting 2 black squares (regardless of the limitations), the first "part" of the calculation is 32....

32 x (something)

Also, since we're asked for the possible ways to choose 2 black squares, that implies that we're dealing with "combinations" of squares, so we'll have to divide by 2 to remove the duplicates.

The ONLY answer that's evenly divisible by 32 is 800; THAT number, when divided by 2 is 400. If I was pressed for time, then I'd have to choose A.

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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Vinay Pathak: Newbie | Next Rank: 10 Posts; Posts: 1; Joined: Sun Mar 04, 2018 10:59 pm

by Vinay Pathak » Sun Mar 04, 2018 11:00 pm

parveen110 wrote:In how many ways is it possible to choose two black squares on a 8 Ã— 8 chessboard so that the squares do not lie in the same row or same column?

a. 400
b. 200
c. 800
d. 100
e. 475

Like this post

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