A company that ships boxes to a total of 12

[guest2]

13. 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

Is this a combinations problem? I can't figure out how to figure this problem out

[Guest]

Since it is a minimum number; let us start with 4:

With 4 colors:

One color at a time: 4 Options

Two Colors at a time: 6 Options.

So with 4 colors we can make 12 possible color codes (or combinations)

You can straight off take 5 as the answer; or try and solve for 5 and see that you can service up to: 5+10 = 15 DCs

[RonPurewal]

http://www.manhattangmat.com/forums/post2399.html

it's much easier just to try the answers than to attempt setting up an equation.
the last two choices are WAY too big - (d) you clearly don't need a separate color for each distribution center, and (e) um, what? are you kidding me?
i think (d) and (e) are probably in there for students who don't really understand the question stem; admittedly, this is not the world's easiest question to understand upon first reading.

if you try 4, you'll get 4 (single colors) + 4!/2!2! (pairs of colors), which is 4 + 6 = 10. this is not enough.
at this point you can just realize that the addition of a fifth color will add significantly more than 2 extra possibilities - you get the fifth color itself, as well as its combination with EVERY one of the existing colors - so it's easy to tell that 5 colors will be good enough without actually computing the total.