The Correct Answer and Explanation is:
Based on a careful mathematical analysis of the problem, none of the provided options is correct. The question is flawed as the correct answer is not listed.
Here is a detailed explanation of the correct calculation:
This problem is best solved using the fundamental counting principle. This principle helps determine the total number of outcomes when there are multiple independent events. In this case, painting each of the four knobs represents an independent event. The choice of color for one knob does not restrict the choice of color for any other knob.
The problem states there are four distinct knobs on the coat hanger and six different colors of paint available.
For the first knob, there are 6 possible color choices.
Since the colors can be reused, there are also 6 possible color choices for the second knob.
Similarly, there are 6 color choices for the third knob.
Finally, there are 6 color choices for the fourth knob.
To find the total number of different ways the knobs can be painted, we multiply the number of choices for each independent event together. The calculation is as follows:
Total Ways = (Choices for Knob 1) × (Choices for Knob 2) × (Choices for Knob 3) × (Choices for Knob 4)
Total Ways = 6 × 6 × 6 × 6 = 6⁴
Total Ways = 1296
The correct number of ways to paint the four knobs is 1,296. This result is not available among the options a. 8, b. 16, c. 24, or d. 36.
The provided options likely stem from common mistakes or typographical errors in the question. For instance, the answer 24 is the result of incorrectly multiplying 6 by 4. The answer 36 would be correct if there were only two knobs instead of four (6² = 36). The answer 16 would be correct if there were four knobs but only two available colors (2⁴ = 16). Given the information in the problem, the question itself is erroneous.
