Final answer:
There are 625 ways to distribute 4 different balls into 5 different boxes, as each ball has 5 choices, making the total number of distributions equal to 5 to the power of 4.
Step-by-step explanation:
The question is concerned with the number of ways 4 different balls can be distributed among 5 different boxes, with any box being able to contain any number of balls.
This situation is analogous to distributing 4 balls into 5 partitions where each partition represents a box. The problem can be solved using the bars and stars method.
There are 5^4 possible distributions because each of the 4 balls can go into any of the 5 boxes.
For example, if the balls are labeled A, B, C, and D and the boxes are labeled 1 through 5, one possible distribution could be A in box 1, B in box 2, C in box 1, and D in box 5.
This is just one of the many combinations possible.
To calculate the total number, consider each ball independently. Ball A has 5 choices, just like balls B, C, and D. Multiplying the number of choices for each ball gives the total number of different distributions:
5 * 5 * 5 * 5 = 5^4 = 625 ways.