OG combinatorics problem

[Redhorsep]

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)

[Anurag@Gurome]

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.

[shayam]

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 ?

[saketk]

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

[shayam]

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

I now understand the reason for nc1 + nc2

[sowmyachada]

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 ??

[abhi131]

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.

[tanviet]

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?