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605-655 (Medium)|   Combinations|                     
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amiteshiitr
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A company that ships boxes to a total of 12 distribution centers uses 22 Feb 2021, 05:03
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Hii,

I have one submission on the answer as 5.
Let take Red and Blue two colours.

Then on the box, RB is not same as BR.

Now let take 4 colours

Now, 4C1 (selecting one colour at a time) + 4C2*2! (Selecting 2 colour at a time X 2! as the colour is different) = 12

Please elaborate why not A option. Thanks!!
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A company that ships boxes to a total of 12 distribution centers uses 22 Feb 2021, 20:01
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Hi amiteshiitr,

The last piece of information in the prompt tells us: "Assume that the order of the colors in a pair does not matter"

When the order does not matter, then RB and BR are the SAME option (so you can't count it twice - you can only count it once). This means that the calculation that you have created is not correct relative to the information that we're given. In these sorts of questions, it can often be fastest to just 'list out' the possibilities (as opposed to doing lots of complex calculations).

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A company that ships boxes to a total of 12 distribution centers uses 01 Jun 2022, 02:27
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EMPOWERgmatRichC ThatDudeKnows JeffTargetTestPrep not clear here why we use >=12 and not =12. After solving, i know 4 cannot be the answer but how did you determine before hand that it should be >=12
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A company that ships boxes to a total of 12 distribution centers uses 01 Jun 2022, 09:51
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Elite097
EMPOWERgmatRichC ThatDudeKnows JeffTargetTestPrep not clear here why we use >=12 and not =12. After solving, i know 4 cannot be the answer but how did you determine before hand that it should be >=12
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If you can do 13, you can do 12. If you can do 14, you can do 12. If you can do 15, you can do 12. If you can do anything greater than 12, you can do 12. That's where people came up with the >=12 bit, but it's not necessary; you're totally fine just solving for =12.

On questions like these, it's often just easier to use brute force than to mess around with the math.
4: A, B, C, D, AB, AC, AD, BC, BD, BD. That's 10. Not enough, but it's pretty close. Adding an E will obviously get us there.

Answer choice B.
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A company that ships boxes to a total of 12 distribution centers uses 20 Jul 2022, 04:56
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Let the number of colours needed to be x.

Since the colours can be either separately used or in pairs,
Total combinations = xC1 + xC2 = x + x(x-1)/2 = (2x + x^2 - x)/2

It is given that the colours are used to paint 12 centres,

(2x + x^2 - x)/2 > 12
(2x + x^2 - x) >24
x^2 + x - 24 > 0

For x = 4, x^2 + x - 24 = -4.
For x = 5, x^2 + x - 24 = 6.

Thus, the minimum number of colours needed for the coding = 5

The correct answer is B.
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A company that ships boxes to a total of 12 distribution centers uses 26 Jul 2023, 10:35
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Say suppose "n" be the no. of colors as per required condition

for single color, no. of flags = n

for double color, no. of flags = n(n-1)/2

Total = n+n(n-1)/2=n(n+1)/2

As per given question n(n+1)/2>12 therefore minimum value n can assume is 5

Hence B
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A company that ships boxes to a total of 12 distribution centers uses 26 Aug 2023, 00:01
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Bunuel
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 the colors in a pair does not matter.)

(A) 4
(B) 5
(C) 6
(D) 12
(E) 24
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In this question if we take into account the fact that the order of the colors in a pair does not matter, then BG(blue and green) and GB( green and blue) can be two unique combinations. With that logic in mind, we can see that 4 colors are sufficient to identify each center. Now, am I missing any understanding here ?
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A company that ships boxes to a total of 12 distribution centers uses 26 Aug 2023, 02:24
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Bunuel
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 the colors in a pair does not matter.)

(A) 4
(B) 5
(C) 6
(D) 12
(E) 24

In this question if we take into account the fact that the order of the colors in a pair does not matter, then BG(blue and green) and GB( green and blue) can be two unique combinations. With that logic in mind, we can see that 4 colors are sufficient to identify each center. Now, am I missing any understanding here ?
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Your interpretation is incorrect. When the question says the order doesn't matter, it means BG and GB are the SAME combination, not two different ones. You've misunderstood the problem's premise. Please reread the question and also go through the solutions in the topic to get a clearer understanding.
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A company that ships boxes to a total of 12 distribution centers uses 03 Sep 2023, 10:43
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A company that ships boxes to a total of 12 distribution centers uses 05 Jan 2026, 16:51
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I solved this using simple counting. We know that there are 12 total centers and each is represented by either a single color or a pair of two different colors. We are given that we can assume that the order of the colors in a pair does not matter. I assigned a letter to each color (A, B, C, etc.) and listed out the potential designations as such. See below:

12 Total Centers

1. A
no more unique designations can be created so we add a new color, 'B'
2. B
3. AB
no more unique designations can be created so we add a new color, 'C'
4. C
5. AC
6. BC
no more unique designations can be created so we add a new color, 'D'
7. D
8. AD
9. BD
10. CD
no more unique designations can be created so we add a new color, 'E'
11. E
12. AE

In total we needed a minimum of 5 colors in order to properly identify each center. Therefore the answer is B.
Bunuel
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 the colors in a pair does not matter.)

(A) 4
(B) 5
(C) 6
(D) 12
(E) 24

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