To solve this, we need to calculate the number of ways to place 9 balls into 9 boxes, where 5 of the balls cannot fit into 3 of the smaller boxes. Let's break this down:

Step 1: Understand the constraints

• There are 9 balls in total.
• There are 9 boxes.
• 5 of the balls cannot be placed in 3 of the smaller boxes.

Step 2: Divide the problem into two groups

• 5 specific balls (that can't go into the 3 small boxes): These 5 balls must be placed into the remaining 6 larger boxes.
• 4 other balls: These can be placed anywhere, including in the 3 smaller boxes.

Step 3: Place the 5 restricted balls into the 6 larger boxes

Since the 5 specific balls cannot go into the 3 smaller boxes, they must go into the remaining 6 boxes. The number of ways to place the 5 balls into the 6 boxes is calculated by choosing 5 boxes out of the 6 and arranging the balls. This can be done as follows:

• Choose 5 boxes out of the 6: $\binom{6}{5}$
• Arrange the 5 balls in those 5 boxes: $5!$

Thus, the number of ways to place the 5 restricted balls into the 6 larger boxes is: $\binom{6}{5} \times 5! = 6 \times 120 = 720$

Step 4: Place the 4 unrestricted balls into the 4 remaining boxes

Once the 5 restricted balls have been placed, the remaining 4 balls can be placed freely into the 4 remaining boxes (including the 3 small boxes). The number of ways to arrange 4 balls in 4 boxes is simply $4!$.

Thus, the number of ways to arrange the 4 unrestricted balls in the 4 remaining boxes is: $4! = 24$

Step 5: Multiply the results

The total number of ways to arrange all the balls is the product of the number of ways to place the 5 restricted balls and the number of ways to place the 4 unrestricted balls: $720 \times 24 = 17,280$

Final Answer:

The number of ways to arrange one ball in each of the boxes is $\boxed{17,280}$.

Do you want further details or have any questions?

Here are 5 related questions for practice:

1. What is the number of ways to arrange 7 balls into 7 boxes if 4 of the balls cannot fit into 2 small boxes?
2. How does the arrangement change if all the balls can fit into any box?
3. How many ways are there to arrange 10 balls into 10 boxes if 6 of them cannot fit into 4 small boxes?
4. What is the significance of using combinations and permutations in this type of problem?
5. Can you calculate the number of ways to arrange 5 balls into 3 boxes without restrictions?

Tip: When faced with restrictions in combinatorial problems, break the problem into smaller parts and handle each part according to the constraints.