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Permutations and Combinations · Study material
Permutations and Combinations Cheat Sheet
Permutations and Combinations is one of the most important and logical topics in Quantitative Aptitude and is frequently asked in SSC, Banking, Railway, UPSC, Defence, Insurance, CAT, and other competitive examinations.
This cheat sheet helps candidates understand counting principles, factorial concepts, arrangement and selection techniques, circular arrangements, team selection problems, and shortcut methods for solving permutation and combination questions quickly and accurately in exams.
About This Cheat Sheet
Complete guide to Permutations and Combinations for competitive exams:
Counting Principles
- Addition Principle
- Multiplication Principle
- Counting Methods
- Arrangement Basics
Factorial Concepts
- n! Definition
- 0! = 1 Rule
- Factorial Simplification
- Factorial Properties
Permutations
- nPr Formula
- Arrangements
- Permutations with Repetition
- Circular Permutations
Combinations
Selection methods using nCr formula and combination techniques.
Word Arrangement Problems
- Letters of Words
- Vowels Together
- Consonants Together
- Identical Object Arrangements
Selection Problems
- Team Selection
- Committee Formation
- Mixed Group Selection
- Distribution Problems
Special Applications
- Polygon Diagonals
- Handshake Problems
- Number Formation
- Digit Arrangement Problems
Shortcut Tricks
Fast solving methods, arrangement shortcuts, and exam-oriented counting techniques.
Topics Covered
| Topic | Description |
|---|---|
| Fundamental Principle of Counting | Addition and multiplication principles |
| Factorial | n! notation and properties |
| Permutation Formula | nPr = n!/(n-r)! |
| Combination Formula | nCr = n!/[r!(n-r)!] |
| Permutation-Combination Relation | nPr = nCr × r! |
| Permutations with Repetition | Repeated arrangements allowed |
| Identical Object Arrangements | Arrangements with repeated objects |
| Circular Permutations | Arrangements in circular form |
| Word Arrangement Problems | Arranging letters of words |
| Selection Problems | Choosing teams and committees |
| Number Formation | Forming numbers using digits |
| Handshake Problems | Combination based handshakes |
| Polygon Diagonals | Diagonal calculations using combinations |
| Shortcut Tricks | Fast solving methods for exams |
Why This Cheat Sheet is Important
Understand Counting Concepts
↓
Improve Logical Thinking
↓
Apply Shortcut Techniques
↓
Solve Arrangement Problems Faster
↓
Score Better in Exams
Competitive Exams Covered
- SSC CGL
- SSC CHSL
- Bank PO & Clerk
- Railway Exams
- UPSC CSAT
- Defence Exams
- Insurance Exams
- State Government Exams
- MBA Entrance Exams
Quick Examples
Example 1:
Find the value of 5!
5! = 5 × 4 × 3 × 2 × 1
= 120
Example 2:
Find the number of permutations of 5 objects taken 2 at a time.
nPr = n!/(n-r)!
= 5!/(5-2)!
= 5!/3!
= 5 × 4
= 20
Example 3:
Find the number of combinations of 6 objects taken 3 at a time.
nCr = n!/[r!(n-r)!]
= 6!/[3! × 3!]
= 20
Example 4:
In how many ways can 4 persons sit around a circular table?
Circular permutation =
(n − 1)!
= (4 − 1)!
= 3!
= 6 ways
Example 5:
How many diagonals can be drawn in a hexagon?
Number of diagonals =
nC2 − n
= 6C2 − 6
= 15 − 6
= 9
Final Takeaway
This Permutations and Combinations Cheat Sheet provides complete revision material for all major concepts frequently asked in competitive examinations. Regular practice of counting principles, arrangement problems, selection methods, and shortcut techniques can significantly improve speed, accuracy, and logical problem-solving ability.
It is highly useful for quick revision before exams and helps candidates strengthen the foundation of Quantitative Aptitude.
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