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

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In a certain city each of 20 Girl Scout troops is represente  [#permalink]

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New post Updated on: 23 May 2014, 12:53
15
00:00
A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

64% (01:49) correct 36% (01:44) wrong based on 199 sessions

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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

Originally posted by GMATSept on 28 Aug 2010, 22:36.
Last edited by Bunuel on 23 May 2014, 12:53, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: Flag Color problem  [#permalink]

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New post 28 Aug 2010, 23:50
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
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Re: Flag Color problem  [#permalink]

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New post 23 May 2014, 12:47
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.
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Re: In a certain city each of 20 Girl Scout troops is represente  [#permalink]

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New post 23 May 2014, 12:59
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


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.

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Re: In a certain city each of 20 Girl Scout troops is represente  [#permalink]

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New post 27 Apr 2015, 07:00
Can you explain this part, please? \(C^2_n\) --> \(\frac{n(n-1)}{2}\)
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Re: In a certain city each of 20 Girl Scout troops is represente  [#permalink]

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New post 27 Apr 2015, 07:08
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Re: In a certain city each of 20 Girl Scout troops is represente  [#permalink]

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New post 27 Apr 2015, 08:00
thanks a lot Bunuel... completely forgot the formula of combination :(
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Re: In a certain city each of 20 Girl Scout troops is represente  [#permalink]

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New post 27 Apr 2015, 11:54
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.
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Re: In a certain city each of 20 Girl Scout troops is represente  [#permalink]

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New post 28 Apr 2015, 06:25
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.
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Re: In a certain city each of 20 Girl Scout troops is represente  [#permalink]

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New post 15 Mar 2017, 16:32
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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Re: In a certain city each of 20 Girl Scout troops is represente  [#permalink]

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New post 16 May 2019, 08:55
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


total pairs sufficient to get 20 flags
6c2+6 ; IMO B
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Re: In a certain city each of 20 Girl Scout troops is represente   [#permalink] 16 May 2019, 08:55
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