🔢 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:

${}^nP_r = \frac{n!}{(n-r)!}$

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

${}^nP_r = n! / (n - r)!$

Example

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

${}^5P_3 = 5! / (5 - 3)! = 60$

Permutation with Repetition Allowed

• Repetition allowed
• Used in passwords, PINs

Formula

$n^r$

Example

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

$10^4 = 10000$

🗣 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

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

${}^9P_4$

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?

${}^5P_3= 5! / (5 - 3)! = 60$

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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