Permutation and combination are fundamental concepts in mathematics that help count and arrange objects. Whether you’re creating passwords, forming teams, or solving probability problems, understanding these concepts is crucial.
Key Differences at a Glance
| Feature | Permutation | Combination |
|---|---|---|
| Order Matters? | Yes | No |
| Formula | nPr = n!/(n-r)! | nCr = n!/r!(n-r)! |
| Example | Arranging letters (ABC ≠ BAC) | Selecting teams (ABC = BAC) |
This 2,000+ word guide will break down both concepts with clear examples, formulas, and practical applications.
Section 1: Permutation Explained
1.1 What is Permutation?
Permutation refers to the arrangement of objects where order matters.
Real-World Examples:
-
Creating passwords (123 ≠ 321)
-
Awarding gold, silver, bronze medals
-
Arranging books on a shelf
1.2 Types of Permutation
a) Without Repetition
Formula:
nPr = n! / (n-r)!
Example:
How many ways can you arrange 3 books out of 5?
5P3 = 5! / (5-3)! = 60 ways
b) With Repetition
Formula:
n^r
Example:
3-digit PIN with repeated numbers:
10^3 = 1,000 combinations
c) Circular Permutation
Formula:
(n-1)!
Example:
Arranging 5 people around a table:
(5-1)! = 24 ways
Section 2: Combination Explained
2.1 What is Combination?
Combination refers to selecting objects where order doesn’t matter.
Real-World Examples:
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Choosing 3 team members out of 10
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Picking lottery numbers (5,10,15 = 15,5,10)
-
Selecting menu items
2.2 Combination Formula
Standard Formula:
nCr = n! / r!(n-r)!
Example:
Selecting 2 fruits out of 5:
5C2 = 5! / 2!(5-2)! = 10 ways
2.3 Special Cases
-
nC0 = 1 (One way to choose nothing)
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nCn = 1 (One way to choose all)
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nCr = nC(n-r) (Choosing 3 out of 5 = leaving 2 out of 5)
Section 3: Key Differences
| Scenario | Permutation (Order Matters) | Combination (Order Doesn’t Matter) |
|---|---|---|
| Passwords | 123 ≠ 321 | Not applicable |
| Teams | ABC (Captain), BAC (Vice-captain) ≠ ABC (Vice-captain), BAC (Captain) | Team {A,B,C} = {B,A,C} |
| Lottery | 5-10-15 ≠ 15-10-5 | 5-10-15 = 15-10-5 (Same ticket) |
Section 4: Real-World Applications
4.1 Computer Science
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Password cracking: Calculating possible combinations
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Data structures: Optimizing sorting algorithms
4.2 Business & Marketing
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A/B testing: Different ad arrangements
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Product bundling: Calculating discount combinations
4.3 Daily Life
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Meal planning: Choosing 3 dishes out of 10
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Travel routes: Finding the fastest path
Section 5: Common Mistakes & Tips
5.1 Frequent Errors
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Confusing nPr and nCr → Ask: “Does order matter?”
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Miscounting repetitions → Use n!/(n-r)! for unique arrangements
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Circular vs. linear → Remember: (n-1)! for circles
5.2 Pro Tips
✔ Use the mnemonic “Permutation = Position” to remember order matters
✔ For probability questions, always calculate total possible outcomes first
✔ Verify with smaller numbers (e.g., test n=3 before n=100)
Section 6: Practice Problems
Beginner Level
-
Permutation: How many 4-letter words can be formed from “MATH”?
Solution: 4P4 = 4! = 24 -
Combination: How many ways to choose 2 toppings from 5 options?
Solution: 5C2 = 10
Advanced Level
-
Circular Permutation: 6 people around a table with 2 fixed seats?
Solution: (6-1)! × 2 = 240 -
Combination with Repetition: How many fruit baskets with 10 fruits (4 types)?
Solution: (10+4-1)C(4-1) = 286
FAQs About Permutation & Combination
❓ Is “combination lock” a misnomer?
✅ Yes! It should be “permutation lock” since order matters.
❓ When to use factorial (!)?
Factorials (n!) count total arrangements of n distinct items.
❓ Can combinations be greater than permutations?
Never. nCr ≤ nPr since order increases possibilities.
❓ How do I know which formula to use?
Ask: “Does rearranging create a new scenario?”
-
Yes → Permutation
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No → Combination
Conclusion & Resources
Mastering permutations and combinations helps in:
✔ Competitive exams (GRE, GMAT)
✔ Data science (probability, statistics)
✔ Daily problem-solving