Permutation & Combination: Basic Formula + JEE Main | Class 11 Maths

Permutation & Combination: Basic Formula + JEE Main

Permutation and Combination Class 11 Mathematics ka ek highly important chapter hai jo JEE Main mein counting-based questions solve karne ke liye fundamental role play karta hai. Is chapter mein sabse important baat formulas ratna nahi, balki ye identify karna hai ki question mein selection ho rahi hai ya arrangement.

Agar order matters, to generally permutation ka concept use hoga. Agar order does not matter, to combination ka concept use hoga. Isi basic difference ko samajhne ke baad Fundamental Principle of Counting, factorial, nPr, nCr, circular arrangements aur restriction-based questions kaafi easy ho jaate hain.

Is article mein hum P&C ke basic formulas ko step-by-step samjhenge aur JEE Main mein question dekhte hi correct method identify karne ki strategy bhi seekhenge.

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1. Fundamental Principle of Counting

Permutation and Combination ka base concept Fundamental Principle of Counting hai.

Agar kisi task ko complete karne ke liye pehle part ke m ways hain aur doosre part ke n ways hain, aur dono choices sequentially perform karni hain, to total ways:

Total ways = m × n

Isse simple language mein yaad rakho:

AND → Multiply

Example

Ek student ke paas:

• 3 shirts
• 2 pants

Har shirt ke saath 2 pants mein se koi bhi choose ki ja sakti hai.

Therefore:

Total outfits = 3 × 2 = 6

Mutually Exclusive Choices

Agar choices alternatives hain aur ek time par sirf ek option choose karna hai, to generally addition principle use hota hai:

Total ways = m + n

Isse yaad rakho:

AND → Multiply

OR → Add

JEE Main mein counting question ko read karte waqt sabse pehle ye identify karna useful hai ki choices sequential hain ya alternatives.

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2. Factorial – Basic Formula

Permutation aur combination dono mein factorial bahut frequently use hota hai.

Positive integer n ka factorial:

n! = n × (n − 1) × (n − 2) × ... × 2 × 1

Examples

5! = 5 × 4 × 3 × 2 × 1 = 120

4! = 4 × 3 × 2 × 1 = 24

Sabse important special result:

0! = 1

Useful Factorial Relation

n! = n × (n − 1)!

For example:

6! = 6 × 5!

JEE Main mein large factorial expressions ko simplify karne ke liye ye relation bahut useful hota hai.

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3. Permutation – Arrangement

Permutation ka use tab hota hai jab order matters.

Agar n objects mein se r objects select karke arrange karne hain, to number of permutations:

ⁿP_r = n!/(n − r)!

Example

5 students mein se 3 students ko ek row mein arrange karna hai.

Yahan order matter karega. For example:

ABC aur BAC different arrangements hain.

Therefore:

⁵P₃ = 5!/(5 − 3)!

= 5!/2!

= 5 × 4 × 3 = 60

So total arrangements:

60

Key Idea

Permutation → Arrangement → Order matters

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4. Combination – Selection

Combination ka use tab hota hai jab order does not matter.

n objects mein se r objects select karne ke ways:

ⁿC_r = n!/[r!(n − r)!]

Example

5 students mein se 3 students ko select karna hai.

Yahan sirf ye matter karta hai ki kaunse 3 students selected hain. Unka order important nahi hai.

⁵C₃ = 5!/(3!2!)

= 10

Agar A, B, C selected hain, to:

ABC = BCA = CAB

Selection same rahegi.

Key Idea

Combination → Selection → Order doesn't matter

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5. Relation Between nPr and nCr

Permutation aur combination ke beech bahut important relation hai:

ⁿP_r = ⁿC_r × r!

Therefore:

ⁿC_r = ⁿP_r/r!

Is Relation ka Logic

Suppose humein n objects mein se r objects ko arrange karna hai.

First, r objects select karo:

ⁿC_r

Ab selected r objects ko arrange karo:

r!

Therefore:

Selection × Arrangement = Permutation

Hence:

ⁿP_r = ⁿC_rr!

Ye relation questions ko convert karne mein bahut useful hai.

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6. Important Properties of Combination

JEE Main mein nCr ki properties calculation ko kaafi fast bana sakti hain.

Symmetry Property

ⁿC_r = ⁿC_n−r

Example:

¹⁰C₃ = ¹⁰C₇

Isliye agar r ki value n/2 se badi hai, to complementary value n−r use karke calculation easy ho sakti hai.

Pascal Identity

ⁿC_r + ⁿC_r−1 = ⁿ⁺¹C_r

Ye identity combinations ke expressions simplify karne mein useful hai.

Important Values

Expression	Value
nC0	1
nC1	n
nCn	1
nCn−1	n

In values ko yaad rakhna direct JEE Main questions mein time save karta hai.

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7. Arrangement of All Objects

Agar n distinct objects ko ek row mein arrange karna hai, to total arrangements:

n!

Example

5 different books ko ek shelf par arrange karna hai:

5! = 120

Therefore total arrangements:

120

Identical Objects

Agar kuch objects identical hain, to simple n! use nahi kar sakte, kyunki identical objects ko interchange karne se new arrangement create nahi hota.

Agar total n objects mein:

• p objects identical hain
• q objects identical hain
• r objects identical hain

to number of distinct arrangements:

n!/(p!q!r!)

Example: LEVEL

Word LEVEL mein total 5 letters hain.

• L → 2 times
• E → 2 times
• V → 1 time

Therefore:

Number of arrangements = 5!/(2!2!)

= 30

Yahan division isliye kiya gaya hai kyunki identical L aur identical E ko interchange karne se new distinct arrangement nahi milta.

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8. Circular Permutation – Basic Formula

Jab objects ko straight line ke bajay circle ke around arrange kiya jata hai, to circular permutation ka concept use hota hai.

n distinct objects around a circle = (n − 1)!

Example

5 people ko circular table ke around arrange karna hai:

(5 − 1)! = 4! = 24

Linear arrangement mein:

n!

Circular arrangement mein:

(n − 1)!

Clockwise/Anticlockwise Mirror Images Same

Agar circular arrangement mein clockwise aur anticlockwise mirror images ko same maana jaye, to:

Number of arrangements = (n − 1)!/2

Important: Circular questions mein pehle carefully identify karo ki rotations aur reflections ko kaise treat kiya ja raha hai.

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9. Restriction-Based Questions

JEE Main mein basic nPr aur nCr ke saath conditions bhi di ja sakti hain, jaise:

• Together
• Not together
• At least
• At most
• Fixed position
• Particular objects cannot be adjacent

In questions mein condition ko formula se pehle samajhna sabse important hai.

Together Method – Block Method

Agar do objects A aur B ko together rakhna hai, to unhe ek single block treat karo:

[AB]

Example: A, B, C, D ko arrange karna hai jahan A aur B together hon.

Objects ban jayenge:

[AB], C, D

Ab total 3 objects hain:

3!

Lekin block ke andar A aur B do ways mein arrange ho sakte hain:

AB, BA → 2!

Therefore:

Total ways = 3! × 2! = 12

Block Method ka Master Idea

Together → Block banao → Blocks arrange karo → Internal arrangements multiply karo.

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10. Selection vs Arrangement – Sabse Important Difference

Feature	Permutation	Combination
Main idea	Arrangement	Selection
Order	Matters	Does not matter
Formula	nPr = n!/(n−r)!	nCr = n!/[r!(n−r)!]
Example	3 students ko row mein arrange karna	3 students ko team mein select karna

One-line trick:

Selection = Combination

Arrangement = Permutation

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11. JEE Main Problem-Solving Method

Permutation and Combination ka question dekhte hi directly formula mat lagao. Pehle following flow follow karo.

Step 1 – Selection ya Arrangement?

Sirf group/team/committee select karni hai → nCr

Positions/order/arrangement create karna hai → nPr

Step 2 – Kya Order Matter Karta Hai?

Yes → Permutation

No → Combination

Step 3 – Special Condition Check Karo

Question mein dekho:

• Repetition allowed hai?
• Kuch objects identical hain?
• Circular arrangement hai?
• Objects together hain?
• Objects not together hain?
• Koi fixed position hai?
• At least/at most condition hai?

Master Flow

Select → nCr

Arrange selected objects → nPr

Arrange all distinct objects → n!

Circular arrangement → (n − 1)!

Together → Block Method

Identical objects → Divide by factorials

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12. JEE Main-Level Solved Question

Question

How many ways can 3 students be selected and arranged in a row from 6 students?

Method 1: Selection + Arrangement

First 6 students mein se 3 students select karo:

⁶C₃ = 20

Ab selected 3 students ko arrange karo:

3! = 6

Therefore:

Total ways = 20 × 6 = 120

Method 2: Direct Permutation

Since final task arrangement hai, directly:

⁶P₃ = 6!/(6 − 3)!

= 6!/3!

= 6 × 5 × 4 = 120

Final Answer: 120

Question ka Main Learning

Question mein “selected and arranged” dono words hain. Isliye pehle selection + arrangement ka logic samajh sakte hain, lekin final answer direct nPr se bhi nikala ja sakta hai.

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13. JEE Main Quick Examples

Example 1 – Selection

8 students mein se 2 students ki team select karni hai.

Order matter nahi karta:

Answer = ⁸C₂

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Example 2 – Arrangement

8 students mein se 2 students ko first aur second position deni hai.

Positions different hain, therefore order matters:

Answer = ⁸P₂

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Example 3 – All Objects

6 different books ko arrange karna hai:

Answer = 6!

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Example 4 – Circular

6 people ko circular table par arrange karna hai:

Answer = (6 − 1)! = 5!

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Example 5 – Together

4 different objects A, B, C, D mein A aur B together hone chahiye.

Aur B ko ek block treat karo:

[AB], C, D

Blocks arrange:

3!

Internal arrangements:

2!

Therefore:

Total = 3! × 2! = 12

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14. Common Mistakes in Permutation & Combination

1. nPr aur nCr ko confuse karna

Sabse common mistake hai formula yaad hone ke baad bhi question ki nature identify na karna.

Order matters → nPr

Order does not matter → nCr

2. Selection ko arrangement samajhna

Team, committee ya group selection mein order generally matter nahi karta, isliye combination concept use hota hai.

3. Circular arrangement mein n! laga dena

Standard circular arrangement of n distinct objects ke liye:

(n − 1)!

4. Identical objects ko distinct treat karna

Agar repeated/identical objects hain, to duplicate arrangements count ho sakte hain. Isliye required factorials se divide karna hota hai.

5. Together condition ko ignore karna

Together condition mein directly n! mat lagao. Pehle block method check karo.

6. “AND” aur “OR” ko confuse karna

Counting principle mein:

AND → Multiply

OR → Add

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15. JEE Main Fast-Solving Strategy

Exam mein calculation se pehle question ko classify karna seekho.

1. Question ka final task identify karo.
2. Selection hai ya arrangement?
3. Order matters?
4. Repetition/identical objects check karo.
5. Circular condition check karo.
6. Together/not together condition check karo.
7. Correct formula apply karo.

Agar question simple selection hai, unnecessary permutation mat lagao.

Agar positions important hain, combination par ruk mat jao.

JEE Main Master Thought:

“Pehle question ko samjho, phir formula lagao.”

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16. Complete Formula Sheet

Concept	Formula / Rule
Fundamental Principle – AND	Multiply
Fundamental Principle – OR	Add
Factorial	n! = n(n−1)!
Special Factorial	0! = 1
Permutation	nPr = n!/(n−r)!
Combination	nCr = n!/[r!(n−r)!]
P-C Relation	nPr = nCrr!
Symmetry	nCr = nCn−r
Pascal Identity	nCr + nCr−1 = n+1Cr
All distinct objects in a row	n!
Identical objects	n!/(p!q!r!...)
Circular arrangement	(n−1)!
Mirror images same	(n−1)!/2
Together condition	Block Method

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17. One-Minute Revision

Exam se pehle P&C ko quickly revise karna ho to sirf ye points yaad rakho:

Fundamental Principle:

AND → Multiply

OR → Add

Factorial:

n! = n(n−1)!

0! = 1

Permutation:

ⁿP_r = n!/(n−r)!

Order matters.

Combination:

ⁿC_r = n!/[r!(n−r)!]

Order does not matter.

Relation:

ⁿP_r = ⁿC_rr!

Linear Arrangement:

n!

Circular Arrangement:

(n−1)!

Identical Objects:

n!/(p!q!...)

Together:

Block Method

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18. Final Revision Box

Situation	Think
Multiple steps / AND	Multiply
Alternative / OR	Add
Selection	nCr
Arrangement	nPr
Arrange all distinct objects	n!
Identical objects	Divide by factorials
Circular arrangement	(n−1)!
Together	Block Method
Order matters?	Permutation
Order doesn't matter?	Combination

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19. Practice Questions

Question 1

How many ways can 4 students be selected from 9 students?

Question 2

How many ways can 4 students be arranged in a row from 9 students?

Question 3

Find the number of distinct arrangements of the letters of the word LEVEL.

Question 4

In how many ways can 6 people be arranged around a circular table?

Question 5

In how many ways can A, B, C, D be arranged if A and B must always be together?

Question 6

Evaluate ¹⁰C₃ using the symmetry property if useful.

Practice Tip: Har question solve karne se pehle ek line likho:

Selection / Arrangement / Circular / Restriction

Ye habit P&C questions mein unnecessary formula selection ki mistakes ko reduce karegi.

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20. PDF Notes

Permutation & Combination ke complete revision notes yahan access karein:

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21. Frequently Asked Questions (FAQs)

Q1. Permutation aur combination mein basic difference kya hai?

Permutation mein order matters, jabki combination mein order does not matter.

Q2. 5 students mein se 3 students ki team banani ho to nPr use hoga ya nCr?

Team selection mein order matter nahi karta, isliye:

⁵C₃

use hoga.

Q3. 5 students mein se 3 ko first, second aur third positions deni hon to?

Positions different hain, isliye order matter karega:

⁵P₃

Q4. 0! ki value kya hoti hai?

0! = 1

Q5. Circular permutation ka basic formula kya hai?

n distinct objects ko circle mein arrange karne ke standard case mein:

(n − 1)!

Q6. Together condition ko kaise solve karte hain?

Required objects ko ek single block treat karke pehle blocks arrange karo, phir block ke andar possible arrangements multiply karo.

Q7. Identical objects ke case mein factorial se divide kyun karte hain?

Identical objects ko interchange karne se koi new distinct arrangement create nahi hota. Isliye duplicate counting ko remove karne ke liye corresponding factorials se divide kiya jata hai.

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22. Final Thoughts

Permutation & Combination ka foundation simple hai, lekin JEE Main mein questions ki difficulty conditions ki wajah se increase hoti hai. Isliye formulas ke saath question classification ki skill develop karna bahut important hai.

Har P&C question ko solve karne se pehle khud se poochho:

“Mujhe select karna hai ya arrange?”

Agar select karna hai:

nCr

Agar arrange karna hai:

nPr

Agar saare distinct objects arrange karne hain:

n!

Agar circle hai:

(n−1)!

Agar objects together hain:

Block Method

Agar identical objects hain:

Required factorials se divide

Sabse important master rule ko hamesha yaad rakho:

Selection → Combination | Arrangement → Permutation | Together → Block | Circle → (n−1)! | Same Objects → Divide by Factorials

In basic concepts ko strong karne ke baad restriction-based arrangements, repeated objects, circular arrangements aur advanced JEE Main counting problems ko systematically tackle karna kaafi easier ho jayega.