BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

P&C

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Senior | Next Rank: 100 Posts
Posts: 91
Joined: Fri Jan 17, 2014 7:34 am
Thanked: 7 times

P&C

by parveen110 » Tue Jun 24, 2014 4:38 am
In how many ways is it possible to choose two black squares on a 8 × 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
Top
Source: — Problem Solving |
User avatar
GMAT Instructor
Posts: 15539
Joined: Tue May 25, 2010 12:04 pm
Location: New York, NY
Thanked: 13060 times
Followed by:1906 members
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 × 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:
Image
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:
Image
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.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Top
User avatar
Legendary Member
Posts: 1100
Joined: Sat May 10, 2014 11:34 pm
Location: New Delhi, India
Thanked: 205 times
Followed by:24 members

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
"GMATinsight"Bhoopendra Singh & Sushma Jha
Most Comprehensive and Affordable Video Course 2000+ CONCEPT Videos and Video Solutions
Whatsapp/Mobile: +91-9999687183 l [email protected]
Contact for One-on-One FREE ONLINE DEMO Class Call/e-mail
Most Efficient and affordable One-On-One Private tutoring fee - US$40-50 per hour
Top

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 16207
Joined: Mon Dec 08, 2008 6:26 pm
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1268 members
GMAT Score:770

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 × 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.
Image
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
Image
Top

GMAT/MBA Expert

User avatar
Elite Legendary Member
Posts: 10392
Joined: Sun Jun 23, 2013 6:38 pm
Location: Palo Alto, CA
Thanked: 2867 times
Followed by:511 members
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]
Image
Top
User avatar
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 × 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
Top
Post new topic Post Reply
• Page 1 of 1