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# An artist is planning on mixing together any number of different color

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Math Expert
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An artist is planning on mixing together any number of different color  [#permalink]

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19 Jul 2017, 05:51
1
7
00:00

Difficulty:

55% (hard)

Question Stats:

53% (02:04) correct 47% (02:52) wrong based on 46 sessions

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An artist is planning on mixing together any number of different colors from her palette. A mixture results as long as the artist combines at least two colors. If the number of possible mixtures is less than 500, what is the greatest number of colors the artist could have in her palette?

(A) 8
(B) 9
(C) 11
(D) 12
(E) 13

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Posts: 7595
Re: An artist is planning on mixing together any number of different color  [#permalink]

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19 Jul 2017, 06:06
1
Bunuel wrote:
An artist is planning on mixing together any number of different colors from her palette. A mixture results as long as the artist combines at least two colors. If the number of possible mixtures is less than 500, what is the greatest number of colors the artist could have in her palette?

(A) 8
(B) 9
(C) 11
(D) 12
(E) 13

Hi..
Say there are n colours..
We can choose 2, 3,4...N out of these

So nC2+nC3+.....+nCn<500....
nC0+nC1+.....+nCn=$$2^n......... nC2+nC3+.....nCn=2^n-1-n$$..
So $$2^n-1-n<500$$..
2^8=256 and 2^9=512..
But 2^9-1-9=502>500..
So n is 8
A
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Re: An artist is planning on mixing together any number of different color  [#permalink]

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19 Jul 2017, 06:09
Bunuel wrote:
An artist is planning on mixing together any number of different colors from her palette. A mixture results as long as the artist combines at least two colors. If the number of possible mixtures is less than 500, what is the greatest number of colors the artist could have in her palette?

(A) 8
(B) 9
(C) 11
(D) 12
(E) 13

$$2^{n-1} < 500$$

$$2^9 = 512$$, so $$2^{9-1} = 2^8 = 256 < 500$$. n = 8. Ans - A.
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An artist is planning on mixing together any number of different color  [#permalink]

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Updated on: 19 Jul 2017, 08:32
let there is n no of colors in the pallet-

now we can select either 2 color or 3 color or 4 color up to n-. i.e

nc2+nc3+nc4+nc5+........+ncn which is equal to 2^n-n-1

2^n-n-1<500
2^n-n<501

now if n= 8 then 2^8-8=248 and if n=9 then 2^9-9=503 which is grater than 501

so n=8
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Originally posted by GTExl on 19 Jul 2017, 06:22.
Last edited by GTExl on 19 Jul 2017, 08:32, edited 2 times in total.
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Re: An artist is planning on mixing together any number of different color  [#permalink]

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19 Jul 2017, 06:49
A different way to look at this: when the artist is choosing colours for her mixture, for each colour, she has two choices: use the colour, or do not use the colour. So if she has n different colours, she can make 2^n different mixtures in total. But that 2^n counts a few things we don't want to count in this question - it counts the 1 selection where she chooses not to use any colours at all, and the n selections where she chooses to use exactly one colour. Those don't count as 'mixtures' here, so the total number of mixtures she can make is:

2^n - n - 1

From here, by substituting, we can find that n=9 gives us just slightly too many possible mixtures, so n=8 is the answer.
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An artist is planning on mixing together any number of different  [#permalink]

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19 Jul 2017, 08:04
According to the value the question :-

Let us consider number of colors to be n
Minimum colors required to make a mixture is 2.
Maximum colors required/can be used to make a mixture is n.

Hence total number of ways to create a mixture is

nC2+nC3+...nCn =<500 (given in question)

Difficult way to solve:

Use hit and trial by using all the given values. You will arrive at the correct answer

Easier way if you know the formula:

nC1+nC2+nC3...nCn = 2^n-1
or
nC2+nC3+nC4...nCn = (2^n)-1-n

Now using the formula

(2^n)-1 -n <500

Solve for n

Smash that A

+1 Kudos if the answer helps
An artist is planning on mixing together any number of different   [#permalink] 19 Jul 2017, 08:04
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