To determine how many different ways the artist can select 4 colors from a set of 6 colors, we use the combination formula:

$C(n, r) = \frac{n!}{r!(n - r)!}$

Where:

• $n$ is the total number of colors (6),
• $r$ is the number of colors to choose (4).

Plugging in the values:

$C(6, 4) = \frac{6!}{4!(6 - 4)!} = \frac{6!}{4! \cdot 2!}$

Calculating the factorials:

$6! = 720,\quad 4! = 24,\quad 2! = 2$

Now, substituting the factorials into the formula:

$C(6, 4) = \frac{720}{24 \cdot 2} = \frac{720}{48} = 15$

Thus, the artist can select the colors in 15 different ways.

Would you like more details or have any questions? Here are some related questions you might consider:

1. How many ways can the artist select all 6 colors?
2. What if the artist can only choose 3 colors instead?
3. How does the selection change if repetition of colors is allowed?
4. Can you explain the difference between combinations and permutations?
5. How would the number of combinations change if there were 8 colors instead of 6?

Tip: When calculating combinations, always remember that the order of selection does not matter!