In a certain city each of 20 Girl Scout troops is represente

Question

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

soumanag · Answer

Hi GMATSept,

The question is basically asking us to find a number N such that N+ NC2  > 20 (since there are 20 different groups)
Its best to back solve. Will save a lot of time.

if N= 5 => N+NC2 = 5 + 5C2 = 5+10 = 15

If N = 6 => N+ NC2 = 6 + 6C2 = 6+15 = 21

as 21 > 20 therefore 6 different colors will be sufficient to give more than 20 different flags.

Answer B

PiyushK · Answer

Checking for each answer choice 
B.6:
if we have 6 colors we can choose pairs in 6C2 ways = 15 and single color flags are 6. Therefore total number of flags = 21.
ANS B.

Bunuel · Answer

Combination approach: Let the # of colors needed is n , then it must be true that n+C^2_n\geq{20} ( C^2_n - # of ways to choose the pair of different colors from n colors when order doesn't matter) --> n+\frac{n(n-1)}{2}\geq{20} --> 2n+n(n-1)\geq{40} --> n(n+1)\geq{40} --> as n is an integer (it represents # of colors) n\geq{6} --> n_{min}=6 . Answer: B. Trial and error approach: If the minimum number of colors needed is 5 then there are 5 single color flags possible PLUS C^2_5=10 two-color flags --> 5+10=15<20 --> not enough for 12 codes; If the minimum number of colors needed is 6 then there are 6 single color flags possible PLUS C^2_6=15 two-color codes --> 6+15=21>20 --> more than enough for 20 flags. Answer: B. Similar questions to practice: each-student-at-a-certain-university-is-given-a-four-charact-151945.html all-of-the-stocks-on-the-over-the-counter-market-are-126630.html if-a-code-word-is-defined-to-be-a-sequence-of-different-126652.html a-4-letter-code-word-consists-of-letters-a-b-and-c-if-the-59065.html a-5-digit-code-consists-of-one-number-digit-chosen-from-132263.html a-company-that-ships-boxes-to-a-total-of-12-distribution-95946.html a-company-plans-to-assign-identification-numbers-to-its-empl-69248.html the-security-gate-at-a-storage-facility-requires-a-five-109932.html all-of-the-bonds-on-a-certain-exchange-are-designated-by-a-150820.html a-local-bank-that-has-15-branches-uses-a-two-digit-code-to-98109.html a-researcher-plans-to-identify-each-participant-in-a-certain-134584.html baker-s-dozen-128782-20.html#p1057502 in-a-certain-appliance-store-each-model-of-television-is-136646.html m04q29-color-coding-70074.html john-has-12-clients-and-he-wants-to-use-color-coding-to-iden-107307.html how-many-4-digit-even-numbers-do-not-use-any-digit-more-than-101874.html a-certain-stock-exchange-designates-each-stock-with-a-85831.html the-simplastic-language-has-only-2-unique-values-and-105845.html m04q29-color-coding-70074.html a-researcher-plans-to-identify-each-participant-in-a-certain-134584-20.html Hope this helps.

evdo · Answer

Can you explain this part, please?  C^2_n  -->  \frac{n(n-1)}{2}

Bunuel · Answer

C^2_n=\frac{n!}{2!*(n-2)!}=\frac{(n-2)!*(n-1)*n}{2!*(n-2)!}=\frac{(n-1)*n}{2}

evdo · Answer

thanks a lot Bunuel... completely forgot the formula of combination

sukanyar · Answer

We have to find the minimum value of the number of colors (n) such that when they are used as 1 color flags or 2 color flags, they are able to represent at least 20 different combinations.

So, C(n,1) + C(n,2) >= 20
n + n(n-1)/2 >= 20
n^2 + n >= 40

So, the minimum value of n that satisfies this is n=6.

ak1802 · Answer

I would say combinatorics is the wrong way to approach this problem, unless you can see that right away you will be plugging in the answer choices.

My approach:

Color 1= 1 total 
Color 2: Color1/2 = 2 total 
Color 3: Color1/3, Color 2/3 = 3 total 
Color 4: Color1/4, Color 2/4, Color 3/4 = 4 total 
Color 5: Color1/5, Color2/5, Color3/5, Color 4/5 = 5 total 
Color 6: ........ 6 total.

Therefore the min number of colors needed is 6.

ScottTargetTestPrep · Answer

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

Archit3110 · Answer

total pairs sufficient to get 20 flags 
6c2+6 ; IMO B

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In a certain city each of 20 Girl Scout troops is represente

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