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

Question

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

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

gmati3 · Accepted Answer

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

Answer is 8.

Smash that A

+1 Kudos if the answer helps

IanStewart · Answer

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.

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.

chetan2u · Answer

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

TimeTraveller · Answer

2^{n-1} < 500

2^9 = 512 , so  2^{9-1} = 2^8 = 256 < 500 . n = 8. Ans - A.

GTExl · Answer

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

Adit_ · Answer

I treated this as a sets problem asking for subsets. We know total subsets of a set = 2^(n)-1 (Subtracting 1 as empty set is not considered)

Now since 2 colors atleast make up the combo, we need to subtract cases with plain 1 color from total which is:

2^n-1 - (Cases with 1)

Now we know 2^9=512 and 2^8=256 (only 2 options that are close rest can be eliminated)

Now if we take 9 as max then 512-1-9=502>500 therefore greatest must be 8.

Answer: Option A IMO

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

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

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