numerology-concept-composition

Complete Guide to Permutations and Combinations

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Probability

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(n, r) = n! / (n – r)!
Example: How many ways to arrange 3 out of 5 books?
P(5,3) = 5! / (2!) = 60

3.2 With Repetition

Each choice can repeat.

Prep(n, r) = nr
Example: 3-digit code using digits 0–9: 103 = 1000 combinations

3.3 Indistinguishable Items (e.g., letters like L, L, O, O, N)

n! / (r1! × r2! × … × rk!)

Where r1, r2… are counts of each repeated item.

“BALLOON” → 7 letters with 2 L’s, 2 O’s:
7! / (2! × 2!) = 1260

3.4 Circular Permutations

Used when arranging around a circle (rotations considered the same).

(n – 1)!
How many ways to arrange 4 people around a round table? (4 – 1)! = 6

4. Combinations (Selections, Order Doesn’t Matter)

4.1 Without Repetition

C(n, r) = n! / (r! × (n – r)!)
Choose 3 students from 6: C(6,3) = 20

4.2 With Repetition

C(n + r – 1, r)

Also called “combinations with replacement”.

Choose 3 fruits from 4 types (can repeat): C(4 + 3 – 1, 3) = C(6,3) = 20

5. Advanced Concept: Buckets and Dividers (Stars and Bars)

When distributing indistinguishable objects into distinguishable bins (e.g. candies to children), we use the stars and bars method.

5.1 Formula

C(r + n – 1, n – 1)
  • r: identical items (stars)
  • n: buckets (dividers)
Distribute 5 candies to 3 kids:
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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