To solve the problem, we need to determine how many different colors can be obtained by mixing any number of paints from a set of five colors: red, blue, green, orange, and white.

Steps to Solve:

1. Understand the Problem:

   • You have 5 colors.
   • You can use any combination of these colors to create a new color.
   • The question asks for the number of distinct colors that can be formed.

2. Combinatorial Analysis:

   • For each color, you have two options: either include it in the mixture or not include it.
   • Since there are 5 colors, the total number of combinations (including the option of using no colors at all) is $2^5$.

3. Subtracting the Null Case:

   • If you choose not to use any color, this results in 0 colors mixed, which isn't a valid mixture.
   • So, you subtract 1 from the total number of combinations.

4. Final Calculation: $\text{Number of colors} = 2^5 - 1 = 32 - 1 = 31$

Conclusion:

You can obtain 31 different colors by mixing any number of the 5 given paints.

Would you like further details or have any questions? Here are some related questions:

1. How would the answer change if there were 6 colors instead of 5?
2. What if each paint could be used in different amounts?
3. Can you explain the difference between combinations and permutations?
4. How would you calculate the number of distinct colors if only specific combinations of paints are allowed?
5. Can you describe how to generalize this approach for $n$ paints?

Tip: When dealing with combinatorial problems, always consider whether the problem allows for repetition or if the order of selection matters. This distinction often affects the calculation method.