Permutations and combinations are two core ideas in combinatorics, used to count how many different ways elements can be arranged or selected. Whether you’re preparing for a math exam, solving a probability puzzle, or working on data science algorithms, understanding these concepts deeply is critical.
1. Basic Principle of Counting
If you can do one task in m ways, and another in n ways, the number of ways of doing both in sequence is m × n. This is known as the multiplication principle.
2. What’s the Difference?
- Permutations – Order matters
- Combinations – Order doesn’t matter
3. Permutations (Ordered Arrangements)
3.1 Without Repetition
Choose r items from n, no repetition, order matters.
P(5,3) = 5! / (2!) = 60
3.2 With Repetition
Each choice can repeat.
3.3 Indistinguishable Items (e.g., letters like L, L, O, O, N)
Where r1, r2… are counts of each repeated item.
7! / (2! × 2!) = 1260
3.4 Circular Permutations
Used when arranging around a circle (rotations considered the same).
4. Combinations (Selections, Order Doesn’t Matter)
4.1 Without Repetition
4.2 With Repetition
Also called “combinations with replacement”.
5. Advanced Concept: Buckets and Dividers (Stars and Bars)
5.1 Formula
- r: identical items (stars)
- n: buckets (dividers)
C(5 + 3 – 1, 3 – 1) = C(7, 2) = 21 ways
5.2 Conditions
If each child must get **at least one**, first give each child one, then distribute the rest using the same formula.
6. When to Use What (Decision Guide)
- Use permutations when order matters
- Use combinations when order doesn’t
- Use nr when repetition is allowed (and order matters)
- Use stars and bars when distributing identical items into groups
- Divide by factorials for indistinct items
7. Real-World Applications
- Seating arrangements at events (permutations)
- Lottery number choices (combinations)
- Password generation (permutations with repetition)
- Inventory distribution problems (stars and bars)
8. Final Words
Permutations and combinations aren’t just about formulas — they’re about logical reasoning. Always ask:
- Does order matter?
- Are items distinct?
- Is repetition allowed?
Mastering these will help you tackle complex counting problems and boost your problem-solving skills in probability, algorithms, and beyond.
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