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

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16 Jun 2010, 09:23

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38% (01:14) wrong based on 791 sessions

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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 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) \(n\geq{5}\) --> \(n_{min}=5\).

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.

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).

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).

Re: A company that ships boxes to a total of 12 distribution [#permalink]

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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) \(n\geq{5}\) --> \(n_{min}=5\).

Answer: B.

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

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) \(n\geq{5}\) --> \(n_{min}=5\).

Answer: B.

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

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
_________________

Re: A company that ships boxes to a total of 12 distribution [#permalink]

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21 May 2017, 08:44

the answer is 5 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

Re: A company that ships boxes to a total of 12 distribution [#permalink]

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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.