Key Concepts and Formulas

• Selections from Identical Objects: The number of ways to select zero or more identical objects from $n$ identical objects is $n+1$.
• Independent Events: If events A and B are independent, the total number of ways both can occur is the product of the number of ways each can occur individually.
• "One or more" Condition: To satisfy the condition of selecting at least one object from a set, subtract 1 from the total number of ways to select zero or more objects, thereby excluding the case where nothing is selected.

Step-by-Step Solution

Step 1: Determine the number of ways to select white balls.

We have 10 identical white balls. We want to find the number of ways to select zero or more white balls. The number of ways to select zero or more white balls is $10 + 1 = 11$. Explanation: We can choose 0 white balls, 1 white ball, 2 white balls, ..., up to 10 white balls. Since the balls are identical, the order doesn't matter, and we are simply counting the number of balls we choose. Thus, there are 11 possibilities.

Step 2: Determine the number of ways to select green balls.

We have 9 identical green balls. We want to find the number of ways to select zero or more green balls. The number of ways to select zero or more green balls is $9 + 1 = 10$. Explanation: Similar to the white balls, we can choose 0 green balls, 1 green ball, 2 green balls, ..., up to 9 green balls. This gives us 10 possibilities.

Step 3: Determine the number of ways to select black balls.

We have 7 identical black balls. We want to find the number of ways to select zero or more black balls. The number of ways to select zero or more black balls is $7 + 1 = 8$. Explanation: We can choose 0 black balls, 1 black ball, 2 black balls, ..., up to 7 black balls. This gives us 8 possibilities.

Step 4: Calculate the total number of ways to select zero or more balls of any color.

Since the selection of balls of each color is independent of the others, we multiply the number of ways for each color to find the total number of ways to select zero or more balls of any color. Total number of ways $= (\text{Ways to select white balls}) \times (\text{Ways to select green balls}) \times (\text{Ways to select black balls})$ Total number of ways $= 11 \times 10 \times 8 = 880$. Explanation: This product gives us all possible combinations of selections, including the case where we select no balls at all.

Step 5: Adjust for the condition "one or more balls".

The problem requires us to select "one or more balls," meaning we must select at least one ball in total. The 880 ways we calculated in Step 4 include the case where we select zero balls of each color. We must exclude this case. Number of ways to select one or more balls $= (\text{Total ways to select zero or more balls}) - 1$ Number of ways to select one or more balls $= 880 - 1 = 879$. Explanation: We subtract 1 to remove the case where we select 0 white balls, 0 green balls, and 0 black balls.

Common Mistakes & Tips

• Always remember to add 1 when counting the number of ways to select from identical objects to account for selecting zero objects.
• Carefully consider the "one or more" condition. It requires subtracting the case of selecting nothing.
• Recognize the independence of selecting balls of different colors, which allows for multiplication.

Summary

We calculated the number of ways to select zero or more balls of each color (white, green, and black) and multiplied these numbers together. Then, to satisfy the condition of selecting at least one ball, we subtracted 1 from the total to exclude the case where no balls are selected. The final answer is 879.

Final Answer

The final answer is \boxed{879}, which corresponds to option (D).