an artist can select 4 colors for a sign. if the artist has 6 colors to choose from how many different ways can he paint the sign

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!

An artist can select 4 colors for a sign. If the artist has 6 colors to choose from, how many different ways can he paint the sign?

Discover how to calculate the number of ways an artist can select 4 colors from a set of 6. This problem demonstrates the use of the combination formula, resulting in 15 unique ways to paint a sign. Ideal for students in Grades 7-9.

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