OG combinatorics problem

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Redhorsep: Senior | Next Rank: 100 Posts; Posts: 49; Joined: Thu Jun 30, 2011 2:47 pm; Thanked: 1 times

by Redhorsep » Tue Sep 20, 2011 6:49 pm

Hi,

Please help me solve this problem from OG quant book, thanks!

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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 cneter 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)

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Source: Beat The GMAT — Problem Solving | Tue Sep 20, 2011 6:49 pm

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Tue Sep 20, 2011 7:51 pm

Solution:
Let the number of colors used be 'n'.
So, nC1 is the number of combinations possible for single color codes.
Similarly nC2 is the number of combinations possible for a pair of color codes.
This implies nC1 + nC2 >= 12.
Check the options then.
If n = 4, 4C1 + 4C2 = 4 + 6 = 10 which is less than 12.
Now, n = 5 gives 5C1 + 5C2 as 5 + 10 = 15 >= 12.

So, 5 is the minimum number of colors required for coding.

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shayam: Junior | Next Rank: 30 Posts; Posts: 21; Joined: Wed Mar 16, 2011 12:31 pmFollowed by: 1 member; Thanked: 3 times

by shayam » Wed Sep 21, 2011 5:57 am

Hi Anurag.

Is there a typo here?
4C2= 12 not 6.

Also, since question is asking us to choose one color OR combination of two color,
Cant we assume that combination of two color will surely result in minimal combination?

Hence Just calculating nC2 >=12
Plugging in 4, We get 4C2 >= 12. Hence 4 colors ?

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saketk: Legendary Member; Posts: 608; Joined: Sun Jun 19, 2011 11:16 amFollowed by: 8 members; Thanked: 37 times

by saketk » Wed Sep 21, 2011 6:56 am

shayam wrote:Hi Anurag.

Is there a typo here?
4C2= 12 not 6.

Also, since question is asking us to choose one color OR combination of two color,
Cant we assume that combination of two color will surely result in minimal combination?

Hence Just calculating nC2 >=12
Plugging in 4, We get 4C2 >= 12. Hence 4 colors ?

Hey 4C2 is 6 and not 12. Please have a look at permutation and combinations formulas. You will easily understand them

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shayam: Junior | Next Rank: 30 Posts; Posts: 21; Joined: Wed Mar 16, 2011 12:31 pmFollowed by: 1 member; Thanked: 3 times

by shayam » Wed Sep 21, 2011 7:57 am

Hey Saket.. I get that now.
I totally missed out bettween permuation and combination..
thanks.

I now understand the reason for nc1 + nc2

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sowmyachada: Newbie | Next Rank: 10 Posts; Posts: 2; Joined: Mon May 24, 2010 12:40 amFollowed by: 2 members

by sowmyachada » Wed Oct 31, 2012 11:08 am

Can you please explain how did the relation nc1+nc2>=12??
i understand how we get nc1 and nc2 but please explain
why is it >=12 ??

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abhi131: Junior | Next Rank: 30 Posts; Posts: 12; Joined: Wed Oct 31, 2012 5:17 pmFollowed by: 2 members; Thanked: 1 times

by abhi131 » Wed Oct 31, 2012 6:12 pm

There are 12 distribution centers, so the number of colors that the company uses should be enough to make at least 12 different combination, or more. Hence >= 12. Choosing 4 colors can only make 10 combinations, and 5 can make more than 12. So 5 has to be the answer.

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tanviet: Legendary Member; Posts: 1404; Joined: Tue May 20, 2008 6:55 pmFollowed by: 2 members; Thanked: 18 times

by tanviet » Wed Oct 31, 2012 8:54 pm

IF THE code is 2 colors, dose the order of color matter?

if order of colors matters, answer is n=4

question is not clear?

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