To solve the problem, we need to find how many distinct ways we can distribute 4 balls (3 indistinguishable white balls and 1 distinguishable black ball) into 4 indistinguishable boxes.
Let's denote the number of white balls in each box by \( a_1 \), \( a_2 \), \( a_3 \), and \( a_4 \), where \( a_i \) represents the number of white balls in box \( i \) and we need to find non-negative integer solutions to the equation:
\[ a_1 + a_2 + a_3 + a_4 = 3 \]
We also have to take into account the placement of the black ball after distributing the white balls.
Step 1: Count the distributions of white balls
First, we will enumerate the different partitions of the number 3, corresponding to how we can allocate 3 indistinguishable white balls into indistinguishable boxes. The partitions of 3 are:
- \( 3 \) (All three white balls in one box)
- \( 2 + 1 \) (Two white balls in one box and one white ball in another)
- \( 1 + 1 + 1 \) (One white ball in each of three boxes)
Now we can list these partitions in a way that clearly defines the placement in indistinguishable boxes:
- Partition \( 3 \): (3)
- Partition \( 2 + 1 \): (2, 1)
- Partition \( 1 + 1 + 1 \): (1, 1, 1)
Step 2: Place the black ball
Now we need to consider how the black ball can be placed in relation to each distribution of the white balls:
-
For the distribution of \( 3 \):
- There is only 1 box with all 3 white balls. When we add the black ball, it can go into this box or one of the remaining (empty) boxes.
- Total configurations = 1 (black in the box) + 3 (black in empty boxes) = 1 way.
-
For the distribution of \( 2 + 1 \):
- We have one box with 2 white balls and one box with 1 white ball.
- Possible locations for the black ball:
- It can go into the box with 2 white balls (1 way)
- It can go into the box with 1 white ball (1 way)
- It can go into one of the empty boxes (2 empty boxes)
- Therefore, total configurations = 1 + 1 + 2 = 4 ways.
-
For the distribution of \( 1 + 1 + 1 \):
- There are 3 boxes, each containing 1 white ball.
- The black ball can go into any of these 3 boxes (there are no empty boxes since each has a white ball).
- Total configurations = 3 (the black ball can go into any of the three boxes) = 3 ways.
Step 3: Sum the possibilities
Finally, we sum all the possible placements derived from each distribution:
\[ 1 + 4 + 3 = 8 \]
Thus, the total number of ways to place 4 balls (3 indistinguishably white and 1 distinguishable black) into 4 indistinguishable boxes is 8.
Therefore, the answer is:
\[ \boxed{8} \]