1. **State the problem:** We have a box with 6 red balls, 4 green balls, and 3 blue balls. We want to find the number of ways to select 3 balls such that each ball is a different color. 2. **Understand the selection condition:** We want 3 balls, each of a different color. Since there are exactly 3 colors, we need to pick exactly one red, one green, and one blue ball. 3. **Calculate the ways to select one ball of each color:** - Number of ways to choose 1 red ball out of 6 is $\binom{6}{1} = 6$. - Number of ways to choose 1 green ball out of 4 is $\binom{4}{1} = 4$. - Number of ways to choose 1 blue ball out of 3 is $\binom{3}{1} = 3$. 4. **Calculate total ways:** Multiply the independent selections: $$6 \times 4 \times 3 = 72$$ 5. **Final answer:** There are $\boxed{72}$ ways to select 3 balls such that all are different colors.