Combinations and Permutations artist Whilst practicing combinations and permutations in my maths class I came across the following problem on a website: > An artist is planning on mixing together any number of different colours from her palette. A mixture results as long as the artist combines at least two colours. If the number of possible mixtures is less than 500, what is the greatest number of colours the artist could have in her palette? If the problem was 2 colours instead of "at least" two colours then I could use $\binom N2$ but this is not the case, so how can I solve this?

In a given mixture, any colour is either in or out of it. The number of possible colour combinations (including an "empty" mixture of no colours) from $n$ colours is $2^n$, but we exclude the zero-colour and one-colour combinations, so $2^n-n-1$ possible mixtures.

Now we need to solve $$2^n-n-1<500$$ $$2^n-n<501$$ We try $n=8$ and get 248 mixtures, less than 501; we try $n=9$ and get 503 mixtures, which is slightly greater than 501. Hence the artist can have at most eight colours on the palette.