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combinatorics problem

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combinatorics problem

by smallsorrow » Mon Sep 22, 2008 6:56 am
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 in the pair does not matter)

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

Not that it took me a long time to read the Q, I also got it wrong. Can anybody explain ? OA follows...
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by mjjking » Mon Sep 22, 2008 7:00 am
the correct answer is 4.
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by smallsorrow » Mon Sep 22, 2008 7:28 am
It is five...

Can anyone tell us why?
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by stop@800 » Mon Sep 22, 2008 7:39 am
smallsorrow wrote:It is five...

Can anyone tell us why?
with 4 colors you can encode 4c1 + 4c2 offices = 4 + 6 = 10
and
with 5 colors you can encode 5c1 + 5c2 offices = 5 + 10 = 15
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by wikimedia » Mon Sep 22, 2008 10:15 am
stop@800 is right.

Answer is 5. Give a code for each color: 0, 1, 2, 3, 4, 5, 6 and so on.

Using 4 colors, we can have the following :
0, 1, 2, 3, 01, 02, 03, 12, 13 and 23 = 10 unique

5 will give 15 unique codes.
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by Scott@TargetTestPrep » Tue Feb 27, 2018 10:49 am
smallsorrow 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 in the 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.

Answer: B

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by Brent@GMATPrepNow » Tue Feb 27, 2018 11:48 am
smallsorrow 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 in the pair does not matter)

A) 4
B) 5
C) 6
D) 12
E) 24
We need to be able to create AT LEAST 12 codes (to represent the 12 countries).

Let's test the options.
Can we get 12 or more color codes with 4 colors?
Let's see . . .

1-color codes = 4 (since there are 4 colors)
2-color codes = We need to choose 2 colors from 4. This can be accomplished in 4C2 ways (using combinations). 4C2 = 6
So, using 4 colors, the total number of color codes we can create = 4 + 6 = 10
We want to create AT LEAST 12 color codes, so we can eliminate answer choice A.

Aside: If anyone is interested, we have a video on calculating combinations (like 4C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789

Can we get 12 or more color codes with 5 colors?
1-color codes = 5 (since there are 5 colors)
2-color codes = We need to choose 2 colors from 5. This can be accomplished in 5C2 ways (using combinations). 5C2 = 10
So, using 5 colors, the total number of color codes we can create = 5 + 10 = 15
Perfect!

The answer is 5 (B)

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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