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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