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Re: Need Help with a Difficult Combinations Question
Originally Posted by
Topher66
Hello, everyone! So I came across this rather difficult practice combinations question on magoosh.com, and while the answer was given, the explanation for it was not. Here is the question (and the answer hidden by a spoiler tag):
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
Would anyone happen to know exactly how to approach this question?
SPOILER: The answer to the question was A (8.)
Let there be n colors in her palette.
The no. of ways to make a mixture of 2 or more colors is nC2 + nC3 ... nCn <= 500.
Now, add nC0 + nC1 = n+1 to both sides. This gives 2^n <500+n+1. Now just find the smallest integer n which solves this problem. Answer comes to (A) 8.
To check, see that 2^9 does not satisfy the inequality.