On problems like this, the easiest solution is just to use the answer choices and count the possibilities. Imagine these options for colors:
R = red
O = orange
Y = yellow
G = green
B = blue
P = purple
(we'll add more if we need them)

For each answer choice, count the options for a single color and for pairs of colors:

A. 4 colors
single color: R, O, Y, G ---> 4 centers
pairs: RO, RY, RG, OY, OG, YG ---> 6 centers
We could also use "4 choose 2" to get 6 ---> (4!/(2!2!))
4 + 6 = 10 centers total. Not enough colors.

B. 5 colors
single color: R, O, Y, G, B ---> 5 centers
At this point, we know that we'll have more pairs than when we had 4 colors, so we can just infer that we'll have over 12, and this will be enough.
Or, we could do "5 choose 2" ---> 5!/(2!3!) = 10
5 + 10 = 15 ---> more than enough for 12 centers.

The answer is B.

AAPL wrote:GMAT Paper Tests

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

OA B

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, here's 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