In how many ways can 4 balls be placed in 4 boxes if neither the balls nor the boxes are distinguishable?

Since the balls and boxes are indistinguishable, the problem involves finding how many ways we can partition 4 identical balls into 4 identical boxes. This is a combinatorial problem and can be solved using the concept of integer partitions.

### Step-by-step solution:
1. We need to partition the integer 4 into at most 4 parts (each part representing the number of balls in each box). The parts must be non-increasing to avoid double-counting the arrangements.
   
2. The possible partitions of 4 (in non-increasing order) are:

   - \( 4 \): All 4 balls in one box.
   - \( 3 + 1 \): 3 balls in one box, 1 in another.
   - \( 2 + 2 \): 2 balls in one box, 2 in another.
   - \( 2 + 1 + 1 \): 2 balls in one box, and 1 ball each in two other boxes.
   - \( 1 + 1 + 1 + 1 \): Each box gets 1 ball.

3. Let's count these partitions:
   - \( 4 \): 1 way.
   - \( 3 + 1 \): 1 way.
   - \( 2 + 2 \): 1 way.
   - \( 2 + 1 + 1 \): 1 way.
   - \( 1 + 1 + 1 + 1 \): 1 way.

4. Summing up these possibilities gives us:

   \[
   1 + 1 + 1 + 1 + 1 = 5
   \]

So, there are **5 ways** to place 4 indistinguishable balls into 4 indistinguishable boxes.

Would you like more details on any part of this solution or have any questions?

Here are some related questions to further your understanding:
1. In how many ways can 5 indistinguishable balls be placed in 3 indistinguishable boxes?
2. What if the balls are distinguishable but the boxes are not? How does the answer change?
3. How many ways can you place 3 indistinguishable balls into 3 distinguishable boxes?
4. How does the solution change if we allow some boxes to be empty?
5. Can you generalize a formula for distributing \( n \) indistinguishable balls into \( m \) indistinguishable boxes?

**Tip**: When dealing with problems involving indistinguishable objects, always consider the partitions of the total number to determine possible arrangements.

Explore the combinatorial problem of placing 4 indistinguishable balls into 4 indistinguishable boxes using integer partitions. Learn how partition theory helps solve this problem with a detailed step-by-step explanation. Suitable for high school students in Grades 9-12.

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