GMATSept wrote:

In a certain city each of 20 Girl Scout troops is represented by a colored flag. Each flag consists of either a single color or a pair of two different colors. If each troop has a different flag, what is the minimum number of colors needed for the flags. (Assume that the order of colors in a pair on a flag does not matter.)

A. 5

B. 6

C. 10

D. 20

E. 40

We can denote our colors with a letter of the alphabet.

A

B, AB

C, AC, BC

D, AD, BD, CD

E, AE, BE, CE, DE

F, AF, BF, CF, DF, EF (we see that we could have stopped at DF, since that was the 20th flag.)

Thus, we see that 6 colors are needed.

Alternate Solution:

Let’s say we need at least n colors to satisfy the requirement. Let’s calculate the number of different flags that can be represented using n colors: The number of single-color flags is n and the number of two-color flags is nC2 = n(n - 1)/2. Therefore, we need:

n + n(n - 1)/2 > 20

2n + n^2 - n > 40

n^2 + n > 40

Looking at the answer choices, we see that n = 6 is the smallest value that satisfies this inequality.

Answer: B

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