Permutation & Combination - Cheat Sheet
Your last-minute revision companion for mastering Permutation & Combination problems in Civil Service and college entrance exams.
Factorials
Formulas
n! = n × (n-1) × (n-2) × ... × 2 × 1
Meaning
n! = factorial of n, multiply all positive integers from n down to 1
Watch out
Remember 0! = 1, not 0
When to use
Foundation for all permutation and combination calculations
Common values
Value
1
Symbol
0!
Quantity
0 factorial
Value
1
Symbol
1!
Quantity
1 factorial
Value
120
Symbol
5!
Quantity
5 factorial
Important facts
- 0! = 1 by definition
- 1! = 1
- Factorials grow extremely rapidly
- Used in both permutations and combinations
Key definitions
Factorial
Example
5! = 5 × 4 × 3 × 2 × 1 = 120
Product of all positive integers less than or equal to a given number
Permutations
Formulas
nPr = n!/(n-r)!
Meaning
n = total objects, r = objects selected, order matters
Watch out
Don't confuse with combinations - permutations care about order
When to use
When arrangement order is important (president, VP, secretary)
Important facts
- Order is important in permutations
- More permutations than combinations for same n and r
- Used for arrangements, rankings, positions
- Keywords: arrange, order, sequence, first/second/third
Key definitions
Permutation
Example
ABC and BAC are different arrangements
Arrangement of objects where order matters
Combinations
Formulas
nCr = n!/[r!(n-r)!]
Meaning
n = total objects, r = objects selected, order doesn't matter
Watch out
Remember the extra r! in denominator compared to permutations
When to use
When selecting groups where order is not important
Important facts
- Order is not important in combinations
- Always fewer combinations than permutations
- Used for selections, groups, teams
- Keywords: select, choose, pick, combination, group
Key definitions
Combination
Example
Selecting 3 people from 10 for a committee
Selection of objects where order doesn't matter
Problem-Solving Strategies
Important facts
- If order matters, use permutation
- If order doesn't matter, use combination
- Read problem carefully for context clues
- Practice identifying keywords
- Set up the formula correctly before calculating
Key definitions
Permutation Keywords
Example
How many ways to arrange 5 books on a shelf?
Arrange, order, sequence, rank, position, first/second/third
Combination Keywords
Example
How many ways to select 3 students from 10?
Select, choose, pick, group, team, committee
Must remember
- 0! = 1 (special case, memorize this)
- Permutation formula: nPr = n!/(n-r)!
- Combination formula: nCr = n!/[r!(n-r)!]
- Order matters = Permutation, Order doesn't matter = Combination
- Permutation keywords: arrange, order, rank, position
- Combination keywords: select, choose, pick, group
- Always check if r ≤ n in your problem
- Combinations have extra r! in denominator vs permutations
- Read problem context carefully to identify which to use
- Practice canceling factorials to simplify calculations
Last minute tips
- Circle keywords in the problem - they tell you permutation vs combination
- For large numbers, cancel common factors in factorials before multiplying
- Double-check your formula setup before calculating
- If stuck, ask 'Does order matter?' - this determines your approach
- Practice common values like 5!, 6!, 7! for quick mental calculation
Comparison tables
Permutation vs Combination
| Aspect | Permutation | Combination |
|---|---|---|
| Order | Matters | Doesn't matter |
| Formula | nPr = n!/(n-r)! | nCr = n!/[r!(n-r)!] |
| Keywords | Arrange, order, rank | Select, choose, pick |
| Result | More arrangements | Fewer selections |
| Example | President, VP, Secretary | Committee members |
Diagrams
Permutation & Combination Concept Map
Visual overview of all major concepts in Permutation & Combination, showing relationships between topics
Problem-Solving Decision Process
Step-by-step decision tree for determining whether to use permutation or combination in word problems
Formula Application Guide
Visual guide showing how permutation and combination formulas are derived and applied