Step-by-step Solution:

Step 1: Analyze the Constraints
Total Balls = 9.
Total Boxes = 9.
Constraint: 5 specific balls are too big to fit into 3 specific small boxes.
Let's categorize them:
• Large Balls (L): 5 (Cannot go into small boxes)
• Small Balls (S): 4 (Can go into any box)
• Small Boxes (SB): 3
• Large Boxes (LB): 6 (9 total - 3 small)

Step 2: Place the Restricted Items (Large Balls)
The 5 Large Balls can only go into the 6 Large Boxes.
Number of ways to choose 5 spots in the 6 Large Boxes and arrange the 5 Large Balls:
\({}^{6}P_5 = 6 \times 5 \times 4 \times 3 \times 2 = 720\).

Step 3: Place the Remaining Items (Small Balls)
After placing the 5 Large Balls, we have:
• Remaining Balls = 4 (All Small)
• Remaining Boxes = 4 (1 Large Box + 3 Small Boxes)
These 4 Small Balls can be arranged in the remaining 4 boxes without restriction.
Ways = \(4! = 4 \times 3 \times 2 \times 1 = 24\).

Step 4: Calculate Total Ways
Total Ways = (Ways for Large Balls) × (Ways for Small Balls)
Total = \(720 \times 24 = 17280\).