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?

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Explanation

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Official Explanation

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Let the number of colors be

n

.

Minimum colors required to make a mixture is 2.

Maximum colors that can be used is

n

.

Total number of ways to create a mixture is

{}_nC_2 + {}_nC_3 + \dots + {}_nC_n \le 500

(given).

Easier way using the formula:

{}_nC_1 + {}_nC_2 + \dots + {}_nC_n = 2^n - 1

, so

{}_nC_2 + {}_nC_3 + \dots + {}_nC_n = 2^n - 1 - n

.…

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What this question tests

Whether you can count arrangements and selections correctly — permutations vs. combinations.

Overcounting: treating a selection as ordered when order does not matter, or forgetting to divide by duplicates.