In how many different ways can six distinct balls be distributed into four distinct boxes, if each box can hold any number of balls including zero?

Introduction / Context:
This is a standard distribution problem where distinct objects are placed into distinct boxes. There is no restriction on how many balls each box may contain, and a box may even remain empty. Such problems are solved by considering the independent choices for each object.

Given Data / Assumptions:

There are 6 distinct balls.

There are 4 distinct boxes.

Each box can hold any number of balls, including zero.

There is no restriction about nonempty boxes.

Concept / Approach:
Each ball can independently be placed into any one of the 4 boxes. Because the balls are distinct, every distinct assignment of balls to boxes is a different arrangement. For each ball we have 4 choices, and these choices multiply because they are independent. Therefore, the number of arrangements is 4^6.

Step-by-Step Solution:
Step 1: Consider the first ball. It has 4 possible boxes.Step 2: The second ball also has 4 choices, regardless of where the first ball went.Step 3: The same reasoning applies to each of the 6 balls.Step 4: By the multiplication principle, the total number of ways = 4 * 4 * 4 * 4 * 4 * 4 = 4^6.Step 5: Compute 4^6 = (4^3)^2 = 64^2 = 4096.

Verification / Alternative check:
We can think of assigning to each ball a code from 1 to 4 representing the chosen box. There are 4 choices for each of the 6 positions in this code, which again gives 4^6 possible codes. This matches the earlier calculation, confirming 4096 as the correct count.

Why Other Options Are Wrong:
2048 equals 2^11 and has no direct connection to the 4^6 structure here. It might come from halving 4096 without justification.
1296 equals 6^4 and may be chosen by mistakenly swapping the roles of balls and boxes and using 6^4 instead of 4^6.
576 is 24^2 and does not come from any correct counting method for this scenario.

Common Pitfalls:
A common mistake is to treat the balls as identical, in which case the counting method is different and more complex. Another error is to try to constrain the number of balls per box, even though the question explicitly allows any number. When each distinct object can go independently into any of several distinct boxes, the correct approach is to raise the number of boxes to the power of the number of objects.

Final Answer:
The total number of distributions is 4096.