A probability problem by Syed Baqir

Let $n$ the number of colors the artist have

The number of possible mixtures for $k$ colors from $n$ colors is ${n \choose k}$

Since a mixture needs at least two colors then then the number of possible mixtures is

${n \choose 2} + {n \choose 3} + {n \choose 4} + {n \choose 5} + \ldots + {n \choose n}$

We have

${n \choose 0} + {n \choose 1} + {n \choose 2} + {n \choose 3} + \ldots + {n \choose n} = 2^n$

Since

${n \choose 2} + {n \choose 3} + {n \choose 4} + {n \choose 5} + \ldots + {n \choose n} < 500$

$2^n - {n \choose 1} - {n \choose 0} < 500$

$2^n - n - 1 < 500$

$2^n - n < 501$

$\boxed{ n = 8 }$