To solve the problem, we can start by recognizing that since green must be one of the colors, we can simplify our selection process.
Iliana has 6 colors in total:
- Green
- Red
- Yellow
- Purple
- Orange
- Blue
Since green is already chosen, we need to select 3 additional colors from the remaining 5 colors (red, yellow, purple, orange, blue).
We can use the combination formula to determine the number of ways to choose 3 colors from 5. The combination formula is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n - r)!} \]
where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.
Here, \( n = 5 \) (the remaining colors) and \( r = 3 \) (the colors we need to choose).
Calculating this gives:
\[ \binom{5}{3} = \frac{5!}{3!(5 - 3)!} = \frac{5!}{3! \cdot 2!} = \frac{5 \times 4}{2 \times 1} = 10 \]
Thus, the number of ways Iliana can pick 4 different colors with green included is:
\[ \boxed{10} \]
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