Final answer:
The artist has 10 different ways to paint the sign, as they can choose one color out of the available 10 colors for each sign.
Step-by-step explanation:
The subject of the student's question involves combinatorics, which is a field of mathematics that deals with counting combinations and permutations of elements. Specifically, the question asks how many different ways an artist can paint a sign, given a certain number of colors to choose from.
In the provided context, if the artist has 10 colors to choose from and can only paint the sign one color, then there simply are 10 different ways to paint the sign. This is because for each color choice, there is one distinct outcome for the color of the sign.
If the artist were painting multiple sections of the sign and could use colors more than once, the possible combinations would increase. However, with the information given, the answer is that there are just 10 different ways to paint the sign with one out of the available 10 colors.