Permutation & Combination - Theory & Concepts
🔀 Permutation & Combination - Complete Theory
Master counting, arrangement, and selection problems!
🎯 Key Difference
Permutation (Order Matters!)
Arrangement of objects where ORDER is important.
Example: ABC, ACB, BAC, BCA, CAB, CBA
All are DIFFERENT permutations!
Symbol: ⁿPᵣ or P(n,r)
Combination (Order Doesn’t Matter!)
Selection of objects where ORDER is NOT important.
Example: ABC, ACB, BAC, BCA, CAB, CBA
All are the SAME combination!
Symbol: ⁿCᵣ or C(n,r) or (n choose r)
Memory Trick:
- Permutation → Position matters
- Combination → Choice/Collection (order irrelevant)
📐 Basic Formulas
Factorial
n! = n × (n-1) × (n-2) × ... × 3 × 2 × 1
0! = 1 (by definition)
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5,040
8! = 40,320
9! = 362,880
10! = 3,628,800
Permutation Formula
ⁿPᵣ = n! / (n-r)!
Where:
n = total number of objects
r = objects to arrange
Example: ⁵P₃ = 5! / (5-3)!
= 5! / 2!
= 120 / 2
= 60
Special Case:
ⁿPₙ = n! (arranging all n objects)
⁵P₅ = 5! = 120
Combination Formula
ⁿCᵣ = n! / [r! × (n-r)!]
Where:
n = total number of objects
r = objects to select
Example: ⁵C₃ = 5! / [3! × 2!]
= 120 / (6 × 2)
= 120 / 12
= 10
Important Properties:
ⁿC₀ = 1
ⁿCₙ = 1
ⁿCᵣ = ⁿCₙ₋ᵣ
ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ
🔄 Relationship Between P & C
ⁿPᵣ = ⁿCᵣ × r!
Permutation = Combination × Arrangement of selected items
Example: ⁵P₃ = ⁵C₃ × 3!
= 10 × 6
= 60 ✓
💡 Solved Examples
Example 1: Basic Permutation
Q: How many 3-digit numbers can be formed using digits 1, 2, 3, 4, 5 without repetition?
Solution:
We need to arrange 3 digits from 5
Order matters (123 ≠ 321)
⁵P₃ = 5! / 2! = 120 / 2 = 60
Answer: 60
Example 2: Basic Combination
Q: In how many ways can 3 students be selected from 5 students?
Solution:
Selection, order doesn't matter
⁵C₃ = 5! / (3! × 2!)
= 120 / (6 × 2)
= 10
Answer: 10
Example 3: Arranging Letters
Q: How many ways can letters of “BOOK” be arranged?
Solution:
Total letters = 4
But O appears 2 times (identical)
Arrangements = 4! / 2!
= 24 / 2
= 12
Answer: 12
Example 4: Circular Permutation
Q: 5 people sitting around circular table. Find number of arrangements.
Solution:
Circular permutation = (n-1)!
= (5-1)!
= 4!
= 24
Answer: 24
Example 5: Selecting with Restriction
Q: From 5 boys and 4 girls, select committee of 3 boys and 2 girls. How many ways?
Solution:
Boys: ⁵C₃ = 5!/(3!×2!) = 10
Girls: ⁴C₂ = 4!/(2!×2!) = 6
Total = 10 × 6 = 60
Answer: 60
Example 6: At Least/At Most
Q: From 6 people, committees of at least 2 can be formed in how many ways?
Solution:
At least 2 = 2 or 3 or 4 or 5 or 6
= ⁶C₂ + ⁶C₃ + ⁶C₄ + ⁶C₅ + ⁶C₆
= 15 + 20 + 15 + 6 + 1
= 57
Shortcut: Total - unwanted
= 2⁶ - ⁶C₀ - ⁶C₁
= 64 - 1 - 6
= 57 ✓
Answer: 57
Example 7: Identical Objects
Q: How many ways to arrange letters of “MISSISSIPPI”?
Solution:
Total = 11 letters
M = 1, I = 4, S = 4, P = 2
Arrangements = 11! / (1! × 4! × 4! × 2!)
= 39,916,800 / (1 × 24 × 24 × 2)
= 34,650
Answer: 34,650
Example 8: Specific Positions
Q: Word “COMPUTER”. How many arrangements start with C and end with R?
Solution:
C _ _ _ _ _ _ R
C and R are fixed
Remaining 6 letters to arrange in middle
= 6!
= 720
Answer: 720
🎯 Important Patterns
Pattern 1: Arrangements of n Items
Total arrangements = n!
But if some are identical:
= n! / (p! × q! × r! × ...)
where p, q, r are frequencies of identical items
Pattern 2: Circular Arrangements
Linear: n!
Circular: (n-1)!
If clockwise = anticlockwise (like beads):
= (n-1)! / 2
Pattern 3: Selecting All vs Some
From n items:
Select all: Only 1 way (ⁿCₙ = 1)
Arrange all: n! ways
Select r: ⁿCᵣ ways
Arrange r: ⁿPᵣ ways
Pattern 4: Vowels/Consonants Together
Treat grouped items as single unit
Example: ORANGE, keep vowels together
Vowels: OAE (treat as 1 unit)
Units: (OAE), R, N, G = 4 units
Arrangements = 4! × 3!
(4! for units, 3! for vowels within)
⚡ Quick Shortcuts
Shortcut 1: ⁿCᵣ Calculation
⁷C₃ = (7 × 6 × 5) / (3 × 2 × 1)
= 210 / 6
= 35
Easier than calculating full factorials!
Shortcut 2: Symmetry Property
ⁿCᵣ = ⁿCₙ₋ᵣ
If calculating ²⁰C₁₈ is hard:
²⁰C₁₈ = ²⁰C₂ = 190 (much easier!)
Shortcut 3: Consecutive Selection
Selecting r consecutive items from n items in circle:
= n ways
Example: 3 consecutive from 10 in circle = 10
Shortcut 4: Divisibility by Specific Digit
To count numbers divisible by 5:
Last digit must be 0 or 5
Fix last digit, arrange remaining
📊 Special Cases
Selecting from Groups
From m men and n women, select:
- r men and s women: ᵐCᵣ × ⁿCₛ
- at least 1 man: Total - (all women)
- exactly 1 woman: ⁿC₁ × ᵐCᵣ₋₁
Distributing Items
Distributing n identical items to r people:
Each gets at least 1: ⁿ⁻¹Cᵣ₋₁
No restriction: ⁿ⁺ʳ⁻¹Cᵣ₋₁
Derangements
Number of ways to arrange n items where no item is in its original position:
≈ n! / e
⚠️ Common Mistakes
❌ Mistake 1: Confusing P & C
Wrong: Using combination when order matters ✗
Right:
Arrangement/Ranking/Code → Permutation ✓
Selection/Committee/Team → Combination ✓
❌ Mistake 2: Identical Items
Wrong: Counting "BOOK" as 4! = 24 ✗
Right: 4!/2! = 12 (two O's are identical) ✓
❌ Mistake 3: Circular ≠ Linear
Wrong: 5 people in circle = 5! ✗
Right: = (5-1)! = 4! ✓
❌ Mistake 4: At Least Calculation
Wrong: Direct addition (tedious, error-prone) ✗
Right: Total - Unwanted (complement!) ✓
❌ Mistake 5: Selection Formula
Wrong: ⁿCᵣ = n! / r! ✗
Right: ⁿCᵣ = n! / (r! × (n-r)!) ✓
📝 Practice Problems
Level 1:
- How many 4-digit numbers from digits 1-9 without repetition?
- In how many ways can 5 books be arranged on shelf?
- From 10 people, select committee of 4. How many ways?
Level 2:
- How many ways to arrange letters of “SUCCESS”?
- 6 people around circular table. Find arrangements.
- From 8 boys and 6 girls, select 3 boys and 2 girls. How many ways?
Level 3:
- Word “EQUATION”. Arrangements starting with vowel?
- 10 points, no 3 collinear. How many triangles?
- Distribute 12 identical chocolates to 5 children, each gets at least 1. How many ways?
🔗 Related Topics
Uses P&C:
- Probability - Counting favorable outcomes
- Number System - Digit arrangement problems
Related:
- Ratio & Proportion - Distribution problems
Practice:
🎯 Continue Your Learning Journey
Master P&C - Ask: Does order matter? 🔀
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