GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 21 Sep 2018, 12:44

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# A company that ships boxes to a total of 12 distribution

Author Message
TAGS:

### Hide Tags

Intern
Joined: 16 Jun 2010
Posts: 17
A company that ships boxes to a total of 12 distribution  [#permalink]

### Show Tags

Updated on: 31 Jul 2012, 13:49
30
00:00

Difficulty:

55% (hard)

Question Stats:

64% (01:06) correct 36% (01:12) wrong based on 849 sessions

### HideShow timer Statistics

A company that ships boxes to a total of 12 distribution centers uses color coding to identify each center. If either a single color or a pair of two different colors is chosen to represent each center and if each center is uniquely represented by that choice of one or two colors, what is the minimum number of colors needed for the coding? (assume that the order of colors in a pair does not matter)

A. 4
B. 5
C. 6
D. 12
E. 24

Originally posted by chintzzz on 16 Jun 2010, 09:23.
Last edited by Bunuel on 31 Jul 2012, 13:49, edited 1 time in total.
Edited the question and added the OA.
Math Expert
Joined: 02 Sep 2009
Posts: 49300
Re: A company that ships boxes to a total of 12 distribution  [#permalink]

### Show Tags

19 Dec 2013, 08:10
2
19
Manager
Joined: 16 Apr 2010
Posts: 202
Re: need help to solve math question  [#permalink]

### Show Tags

16 Jun 2010, 10:12
7
5

This is a combination problem that can be formulated in the following matter:
XC1 + XC2 >= 12

Lets try answer 4. We will get 4C1+4C2= 4+6 which is less than 10.
Trying 5, we will get 5C1+5C2= 5+10 which is greater than 12,

##### General Discussion
Math Expert
Joined: 02 Sep 2009
Posts: 49300
Re: need help to solve math question  [#permalink]

### Show Tags

16 Jun 2010, 10:20
2
3
chintzzz wrote:
A company that ships boxes to a total of 12 distribution centers uses color coding to identify each center. If either a single color or a pair of two different colors is chosen to represent each center and if each center is uniquely represented by that choice of one or two colors, what is the minimum number of colors needed for the coding? (assume that the order of colors in a pair does not matter)
A)4
B)5
C)6
D)12
E)24

You can solve by trial and error or use algebra.

Let # of colors needed be $$n$$, then it must be true that $$n+C^2_n\geq{12}$$ ($$C^2_n$$ - # of ways to choose the pair of different colors from $$n$$ colors when order doesn't matter) --> $$n+\frac{n(n-1)}{2}\geq{12}$$ --> $$2n+n(n-1)\geq{24}$$ --> $$n(n+1)\geq{24}$$ --> as $$n$$ is an integer (it represents # of colors) $$n\geq{5}$$ --> $$n_{min}=5$$.

Hope it's clear.
_________________
Intern
Joined: 12 Dec 2013
Posts: 3
Re: A company that ships boxes to a total of 12 distribution  [#permalink]

### Show Tags

19 Dec 2013, 06:40
2
I tried to do it with writing the possibilities out:

B(Blue) R(Red) Y(Yellow) P(Pink)

B
BR
RB
R
Y
YR
YB
RY
BY
P
PR
RP
PB
BP
PY
YP

I already reach 16 different combinations with only 4 colours, but the OA is 5? what´s my mistake?

EDIT:

Just figured that the ordering does not count as 2 different orders.. therefore, we need 5 colours.. thanks anyway
Manager
Joined: 06 Dec 2014
Posts: 66
GMAT 1: 670 Q48 V34
Re: A company that ships boxes to a total of 12 distribution  [#permalink]

### Show Tags

18 Feb 2015, 15:23
Can anyone tell me when we use n^2 and when we use n+nC2 ????

In this color question we use n+nC2 >= 12

In integer questions we use n^2>=15 ....
EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 12425
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: A company that ships boxes to a total of 12 distribution  [#permalink]

### Show Tags

18 Feb 2015, 22:24
1
2
Hi gmathopeful90,

The "restrictions" in the question are what dictate the math.

Consider these possible scenarios:

1) You have 5 different colors to choose from and two different rooms to paint. You can use the same color in both rooms. How many different color combinations are there for the two rooms?

Here, the first room could be 5 different colors and the second room could be 5 different colors, so (5)(5) = 5^2 = 25 options.

2) You have 5 different colors to choose from and two different rooms to paint. You CANNOT use the same color in both rooms. How many different color combinations are there for the two rooms?

Here, the first room could be 5 different colors; once you assign that first color, the second room could only be 4 different colors, so (5)(4) = 20 options.

3) You have 5 different colors to choose from. How many different 1-color and 2-color codes can you form with the following restrictions: the 2-color codes must use 2 DIFFERENT colors and the order of the colors does not matter (so blue-green is the SAME code as green-blue)?

Here, you start with the 5 different 1-color codes, then 5c2 different 2-color codes = 5 + 10 = 15 codes.

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

760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

# Rich Cohen

Co-Founder & GMAT Assassin

Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ ***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*********************** Manager Joined: 06 Dec 2014 Posts: 66 GMAT 1: 670 Q48 V34 Re: A company that ships boxes to a total of 12 distribution [#permalink] ### Show Tags 19 Feb 2015, 13:59 Thanks for the reply Do you mean in questions where we assume order of colors in cominations matters, we can use 5*4.. But where color doesn't matter, we use 5C2 ??? This explains stuff for me EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 12425 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: A company that ships boxes to a total of 12 distribution [#permalink] ### Show Tags 19 Feb 2015, 14:24 1 2 Hi gmathopeful90, You've hit on THE key difference between Permutation and Combination questions: does the order MATTER or not. IF you're putting things in order (the word "arrange" or "arrangements" often shows up in these types of questions), then you have to keep track of the number of options at each "step" and standard multiplication is involved. IF you're picking combinations of things (the word "combination" is the common word in these questions), then the order of the items does NOT matter and you have to use the Combination Formula. One of the interesting "design elements" of Official GMAT questions is that you can use either of the above approaches on certain types of prompts - you just have to be careful about how you set up the math (and you have to be really organized with your work). GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************

Manager
Joined: 28 Apr 2016
Posts: 97
Re: A company that ships boxes to a total of 12 distribution  [#permalink]

### Show Tags

12 Jul 2016, 21:14
I am confused with the line 'order doesn't matter'.

Does it mean BR = RB or BR =/ to RB?.

ij78cp wrote:
I tried to do it with writing the possibilities out:

B(Blue) R(Red) Y(Yellow) P(Pink)

B
BR
RB
R
Y
YR
YB
RY
BY
P
PR
RP
PB
BP
PY
YP

I already reach 16 different combinations with only 4 colours, but the OA is 5? what´s my mistake?

EDIT:

Just figured that the ordering does not count as 2 different orders.. therefore, we need 5 colours.. thanks anyway
EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 12425
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: A company that ships boxes to a total of 12 distribution  [#permalink]

### Show Tags

14 Jul 2016, 07:06
Hi ameyaprabhu,

When the order doesn't matter, RB and BR are the SAME option (so you can't count it twice, you can only count it once). In these sorts of questions, it can often be fastest to just 'list out' the possibilities (as opposed to doing lots of complex calculations).

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

760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

# Rich Cohen

Co-Founder & GMAT Assassin

Special Offer: Save \$75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************

Intern
Joined: 27 Oct 2016
Posts: 3
Location: Argentina
Concentration: General Management, Operations
WE: Engineering (Energy and Utilities)
Re: A company that ships boxes to a total of 12 distribution  [#permalink]

### Show Tags

22 Feb 2017, 02:41
Hi writing the possibilities should be like this:

1-color code: A, B, C, D, E --> 5 POS.
2-color code: AB, AC, BC, DA, DB, DC, EA, EB, EC, ED --> 10 POS.

ij78cp wrote:
I tried to do it with writing the possibilities out:

B(Blue) R(Red) Y(Yellow) P(Pink)

B
BR
RB
R
Y
YR
YB
RY
BY
P
PR
RP
PB
BP
PY
YP

I already reach 16 different combinations with only 4 colours, but the OA is 5? what´s my mistake?

EDIT:

Just figured that the ordering does not count as 2 different orders.. therefore, we need 5 colours.. thanks anyway
Intern
Joined: 07 Apr 2017
Posts: 5
Re: A company that ships boxes to a total of 12 distribution  [#permalink]

### Show Tags

24 Apr 2017, 23:50
Bunuel wrote:
chintzzz wrote:
A company that ships boxes to a total of 12 distribution centers uses color coding to identify each center. If either a single color or a pair of two different colors is chosen to represent each center and if each center is uniquely represented by that choice of one or two colors, what is the minimum number of colors needed for the coding? (assume that the order of colors in a pair does not matter)
A)4
B)5
C)6
D)12
E)24

You can solve by trial and error or use algebra.

Let # of colors needed be $$n$$, then it must be true that $$n+C^2_n\geq{12}$$ ($$C^2_n$$ - # of ways to choose the pair of different colors from $$n$$ colors when order doesn't matter) --> $$n+\frac{n(n-1)}{2}\geq{12}$$ --> $$2n+n(n-1)\geq{24}$$ --> $$n(n+1)\geq{24}$$ --> as $$n$$ is an integer (it represents # of colors) $$n\geq{5}$$ --> $$n_{min}=5$$.

Hope it's clear.

Could you please explain me how you get [fraction]n(n-1)/2 from C^2_n? Shouldn't it be [fraction]n!/k!(n-k)! ?
Thanks
Math Expert
Joined: 02 Sep 2009
Posts: 49300
Re: A company that ships boxes to a total of 12 distribution  [#permalink]

### Show Tags

25 Apr 2017, 02:04
matteogr wrote:
Bunuel wrote:
chintzzz wrote:
A company that ships boxes to a total of 12 distribution centers uses color coding to identify each center. If either a single color or a pair of two different colors is chosen to represent each center and if each center is uniquely represented by that choice of one or two colors, what is the minimum number of colors needed for the coding? (assume that the order of colors in a pair does not matter)
A)4
B)5
C)6
D)12
E)24

You can solve by trial and error or use algebra.

Let # of colors needed be $$n$$, then it must be true that $$n+C^2_n\geq{12}$$ ($$C^2_n$$ - # of ways to choose the pair of different colors from $$n$$ colors when order doesn't matter) --> $$n+\frac{n(n-1)}{2}\geq{12}$$ --> $$2n+n(n-1)\geq{24}$$ --> $$n(n+1)\geq{24}$$ --> as $$n$$ is an integer (it represents # of colors) $$n\geq{5}$$ --> $$n_{min}=5$$.

Hope it's clear.

Could you please explain me how you get n(n-1)/2 from C^2_n? Shouldn't it be n!/k!(n-k)! ?
Thanks

$$C^2_n=\frac{n!}{(n-2)!*2!}=\frac{(n-2)!*(n-1)*n}{(n-2)!*2!}=\frac{(n-1)*n}{2}$$.

Hope it's clear.
_________________
Director
Joined: 17 Dec 2012
Posts: 636
Location: India
Re: A company that ships boxes to a total of 12 distribution  [#permalink]

### Show Tags

20 May 2017, 20:13
2
Top Contributor
1
chintzzz wrote:
A company that ships boxes to a total of 12 distribution centers uses color coding to identify each center. If either a single color or a pair of two different colors is chosen to represent each center and if each center is uniquely represented by that choice of one or two colors, what is the minimum number of colors needed for the coding? (assume that the order of colors in a pair does not matter)

A. 4
B. 5
C. 6
D. 12
E. 24

1. Solving a simple case and then generalizing would be easy for this problem.
2. Take 2 colors Red and Blue. These two can be used in the following ways R, B, RB. i.e, 2+2C2. It can represent only 3 centers
3. Take 3 colors R, B, G. These can represent 3 +3c2=6 centers
4. Four colors can represent 4+4C2= 10 centers
5 colors can represent 5+5C2=15 centers

So we see a minimum of 5 colors are needed
_________________

Srinivasan Vaidyaraman
Sravna Holistic Solutions
http://www.sravnatestprep.com

Holistic and Systematic Approach

Senior Manager
Joined: 24 Oct 2016
Posts: 285
Location: India
Schools: IIMB
GMAT 1: 550 Q42 V28
GPA: 3.96
WE: Human Resources (Retail Banking)
Re: A company that ships boxes to a total of 12 distribution  [#permalink]

### Show Tags

21 May 2017, 08:44
see if we have 4 different colors then we have 4 unique single identity and 4*3/1*2=6 unique identity with pairs so slightly more than 4 will be the answer that is 5
Manager
Joined: 23 Dec 2013
Posts: 181
Location: United States (CA)
GMAT 1: 710 Q45 V41
GMAT 2: 760 Q49 V44
GPA: 3.76
Re: A company that ships boxes to a total of 12 distribution  [#permalink]

### Show Tags

02 Jun 2017, 11:11
chintzzz wrote:
A company that ships boxes to a total of 12 distribution centers uses color coding to identify each center. If either a single color or a pair of two different colors is chosen to represent each center and if each center is uniquely represented by that choice of one or two colors, what is the minimum number of colors needed for the coding? (assume that the order of colors in a pair does not matter)

A. 4
B. 5
C. 6
D. 12
E. 24

For this problem, you can simply list out the possibilities. Since it's a min/max problem, starting with A is best. Repetition of colors is not allowed.

A) n = 4

Let ABCD represent four colors.

ABCD = 4 centers covered
AB AC AD = 3 more centers
BC BD = 2 more centers
CD = 1 more center

The total here is 11. Since we are close to 12, an increase in one color should be more than enough. B is the answer.
Intern
Joined: 27 Nov 2016
Posts: 5
Re: A company that ships boxes to a total of 12 distribution  [#permalink]

### Show Tags

02 Jun 2017, 12:11
Solve this using algebra

nC2 + nC1 >= 12

n*(n-1)/2 + n >= 12

Testing the answer options , 5 is the minimum value which satisfies the condition.

Sent from my iPhone using GMAT Club Forum
Target Test Prep Representative
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2835
Re: A company that ships boxes to a total of 12 distribution  [#permalink]

### Show Tags

08 Mar 2018, 17:48
chintzzz wrote:
A company that ships boxes to a total of 12 distribution centers uses color coding to identify each center. If either a single color or a pair of two different colors is chosen to represent each center and if each center is uniquely represented by that choice of one or two colors, what is the minimum number of colors needed for the coding? (assume that the order of colors in a pair does not matter)

A. 4
B. 5
C. 6
D. 12
E. 24

Since we have only 12 distribution centers, we know we will need fewer than 12 different colors to identify them.

Let’s say we have 4 different colors; then 4C1 = 4 centers can be identified by one color, and 4C2 = 6 centers can be identified by two different colors. So a total of 4 + 6 = 10 centers can be identified.

We see that if we have only 4 different colors, we don’t have enough ID codes to assign to the 12 centers. Therefore, we need one more color.

If we have 5 different colors, then 5C1 = 5 centers can be identified by one color, and 5C2 = 10 centers can be identified by two different colors. So a total of 5 + 10 = 15 centers can be identified.

We see that if we have 5 different colors, we have more than enough ID codes to assign to the 12 centers.

_________________

Jeffery Miller

GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions

Re: A company that ships boxes to a total of 12 distribution &nbs [#permalink] 08 Mar 2018, 17:48
Display posts from previous: Sort by

# Events & Promotions

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.