🔢 Permutation

Last Updated: Jan 2026

Permutation deals with arrangements where order matters. In exams and programming problems, permutation questions appear in many different forms, not just the basic formula.

🗣 Hinglish Tip: Permutation = arrange karna, position badli to answer badal gaya

Core Formula

Arrangement of r objects from n distinct objects:

ⁿPᵣ = n! / (n − r)!

Variety of Permutation Based on Conditions

Simple Permutation (All Objects Distinct)

  • All objects are different
  • No restriction

Formula

n!

Example

How many ways to arrange letters of ABC?

3! = 6

Permutation of r Objects from n

  • Selecting some objects
  • Order matters

Formula

ⁿPᵣ = n! / (n − r)!

Example

How many 3-letter words can be formed from 5 letters?

⁵P₃ = 60

Permutation with Repetition Allowed

  • Repetition allowed
  • Used in passwords, PINs

Formula

nʳ

Example

How many 4-digit PINs using digits 0–9?

10⁴ = 10000

🗣 Hinglish Tip: Repetition allowed = power formula

Permutation of Objects Not All Distinct

(Repetition of Similar Objects)

  • Some objects are identical

Formula

n! / (p! q! r!)

Where p, q, r are repetitions.

Example (Very Important)

How many arrangements of letters in BANANA?

  • Total letters = 6
  • A repeated 3 times
  • N repeated 2 times
6! / (3! 2!) = 60

Circular Permutation

  • Objects arranged in a circle
  • Rotations are considered same

Formula

(n − 1)!

Example

How many ways can 5 people sit around a round table?

(5 − 1)! = 24

🗣 Hinglish Tip: Circle me rotation same hota hai, isliye ek kam factorial

Permutation with Restrictions

(Conditional Arrangement)

  • Words like:
    • together
    • not together

Case A: Objects Together

Example: In how many ways can A and B sit together among 5 people?

Step-wise Idea:

  • Treat (AB) as one unit
  • Then multiply by internal arrangements

Structure

StepLogic
1Treat restricted objects as one block
2Arrange blocks
3Multiply by internal permutations

Case B: Objects Not Together

Formula:

Total arrangements − Together arrangements

Permutation with Fixed Positions

  • Some positions are fixed

Example

How many 5-digit numbers can be formed if first digit is 5?

  • First digit fixed → remaining 4 digits
⁹P₄

Number Formation Problems

Common Rules

  • First digit ≠ 0
  • Digits not repeated (unless mentioned)
  • Order matters → Permutation

Example

How many 3-digit numbers can be formed using digits 1–5 without repetition?

⁵P₃ = 60

Permutation with Even / Odd Conditions

Example

How many 4-digit numbers formed from digits 1–9 are even?

Logic:

  • Last digit must be even
  • Remaining digits arranged

This type is very common in exams.

Alphabet / Word Arrangement Questions

Keywords:

  • Letters
  • Words
  • Meaningful / meaningless
  • Repetition

Always check:

  • Are letters repeated?
  • Any restriction?
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