Distribution Problem
Distribution problems involve distributing objects into boxes or recipients. When objects are distinct and boxes are distinct, each object has independent choices, so the number of ways is (number of boxes)^(number of objects). Different variations include empty boxes allowed or not, identical objects, or identical boxes.
What You'll Learn
Introduction to Distribution Problem
Distribution problems involve distributing objects into boxes or recipients. When objects are distinct and boxes are distinct, each object has independent choices, so the number of ways is (number of boxes)^(number of objects). Different variations include empty boxes allowed or not, identical objects, or identical boxes.
Prerequisites
How to Solve Distribution Problem Problems
Step 1: Identify if objects are distinct or identical
Step 2: Identify if boxes/recipients are distinct or identical
Step 3: Determine if empty boxes are allowed
Step 4: For distinct objects, distinct boxes, empty allowed: use kⁿ
Step 5: For distinct objects, distinct boxes, no empty: use inclusion-exclusion
Step 6: For identical objects, distinct boxes: use stars and bars
Step 7: For identical objects, identical boxes: use partition theory (advanced)
Example Problem
Example: In how many ways can 5 different gifts be distributed among 3 children? (Empty allowed) Solution: Step 1: Gifts are distinct, children are distinct Step 2: Empty boxes allowed Step 3: Each gift can go to any of 3 children Step 4: Total ways = 3⁵ = 243 Answer: 243 ways
Pro Tips & Tricks
- Distinct objects, distinct boxes, empty allowed: kⁿ
- Distinct objects, distinct boxes, no empty: k! × S(n,k) (Stirling number)
- Identical objects, distinct boxes, empty allowed: C(n+k-1, k-1) (stars and bars)
- Identical objects, distinct boxes, no empty: C(n-1, k-1)
- When distributing distinct objects, each object is independent
- When distributing identical objects, use combinations not permutations
Shortcut Methods to Solve Faster
Common Mistakes to Avoid
Practice Worksheets
Practice makes perfect! Work through these worksheets to master Distribution Problem. Each worksheet contains 20 questions with detailed explanations. Start from Worksheet 1 and progress through increasing difficulty levels.
Exam Importance
Distribution Problem is an important topic for various competitive exams. Here's how frequently it appears:
Ready to Master Distribution Problem?
Start with Worksheet 1 and work your way up to expert level! Each worksheet includes: