Kya Class 11 mein Permutations aur Combinations samajh nahi aate? Chinta mat karo! Aaj main tumhe bilkul pehli baar se samjhaungi — itne simple tarike se ki sab crystal clear ho jaye.
Sabse Pehle — Yeh Topic Kyun Zaroori Hai
Socho tumhare paas 3 kapde hain — aur tumhe 2 kapde choose karne hain alag alag tarike se pehenne ke liye — kitne tarike hain?
Ya socho — 5 dost hain — 3 ko ek saath baithana hai — kitne tarike se baith sakte hain?
Yeh sab Permutations aur Combinations se solve hota hai.
Factorial Kya Hota Hai — Pehle Samjho
Permutations aur Combinations mein Factorial bahut use hota hai.
Factorial — n! matlab:
n! = n × (n-1) × (n-2) × … × 2 × 1
Jaise:
•5! = 5 × 4 × 3 × 2 × 1 = 120
•4! = 4 × 3 × 2 × 1 = 24
•3! = 3 × 2 × 1 = 6
•2! = 2 × 1 = 2
•1! = 1
•0! = 1 — yeh yaad rakhna zaroori hai.
Fundamental Principle of Counting
Yeh sabse basic rule hai — sab kuch isi se shuru hota hai.
Rule 1 — Multiplication Principle:
Agar ek kaam m tarike se ho sakta hai
Aur doosra kaam n tarike se ho sakta hai
To dono saath m × n tarike se honge.
Example:
Shirt ke 3 options hain — pant ke 4 options hain
Total combinations = 3 × 4 = 12 ✅
Rule 2 — Addition Principle:
Agar ek kaam m tarike se ho sakta hai
Ya doosra kaam n tarike se ho sakta hai — dono ek saath nahi
To total = m + n tarike.
Example:
Bus se jaana — 3 routes
Train se jaana — 2 routes
Total ways = 3 + 2 = 5 ✅
Permutation Kya Hota Hai
Permutation — jab order matter karta hai.
Simple matlab — alag alag arrangements.
Jaise — A, B, C ko arrange karo:
•ABC, ACB, BAC, BCA, CAB, CBA — 6 arrangements.
Yahan order matter karta hai — ABC aur BAC alag hain.
Permutation Formula:
ⁿPr = n! / (n-r)!
Jahan:
•n — total items
•r — kitne choose karne hain
Example:
5 logon mein se 3 ko line mein khada karna hai
⁵P3 = 5! / (5-3)! = 5!/2! = 120/2 = 60 ✅
Permutation Ke Khaas Cases
Case 1 — Sab arrange karo:
n items ko arrange karo — n! tarike se.
Jaise — 4 log — 4! = 24 tarike se baith sakte hain.
Case 2 — Repetition allowed:
n items mein se r choose karo — repetition allowed
nʳ tarike se.
Jaise — 4 digit ka PIN — 0-9 mein se — repetition allowed
10⁴ = 10000 PINs possible.
Case 3 — Circular Permutation:
Gol mez pe baithna — circular arrangement
n logon ke liye — (n-1)! tarike.
Jaise — 5 log gol mez pe — (5-1)! = 4! = 24 tarike.
Combination Kya Hota Hai
Combination — jab order matter nahi karta.
Simple matlab — sirf selection — arrangement nahi.
Jaise — A, B, C mein se 2 choose karo:
•AB, AC, BC — sirf 3 combinations.
•AB aur BA same hain — kyunki order matter nahi.
Combination Formula:
ⁿCr = n! / [r! × (n-r)!]
Example:
5 logon mein se 3 ki team banani hai
⁵C3 = 5! / (3! × 2!) = 120 / (6×2) = 10 ✅
Permutation aur Combination Mein Fark
Yeh sabse zaroori concept hai — yaad rakho.
Real Life Examples:
Permutation — order matter karta hai:
•Password — 1234 aur 4321 alag hain ✅
•Race mein position — 1st, 2nd, 3rd — order matter karta hai ✅
•Lock combination — order matter karta hai ✅
Combination — order matter nahi:
•Cricket team — 11 players — order matter nahi ✅
•Pizza toppings — cheese aur mushroom — order matter nahi ✅
•Committee banana — order matter nahi ✅
Zaroori Properties — Pakke Yaad Karo
ⁿCr = ⁿC(n-r)
Jaise — ¹⁰C3 = ¹⁰C7 ✅
ⁿC0 = ⁿCn = 1
Kuch nahi choose karna ya sab choose karna — sirf ek tarika.
ⁿC1 = n
Ek item choose karna — n tarike.
ⁿCr + ⁿC(r-1) = (n+1)Cr
Yeh Pascal’s Triangle se aata hai.
ⁿPr = ⁿCr × r!
Permutation = Combination × Factorial of r
Examples — Step by Step
Example 1 — Permutation:
Ek library mein 6 alag books hain
4 books ko shelf pe arrange karna hai
Kitne tarike se arrange kar sakte hain?
⁶P4 = 6!/(6-4)! = 6!/2! = 720/2 = 360 tarike ✅
Example 2 — Combination:
8 students mein se 3 ki committee banani hai
Kitne tarike se ban sakti hai?
⁸C3 = 8!/(3! × 5!) = 336/6 = 56 tarike ✅
Example 3 — Mixed:
10 logon mein se 5 ki team banani hai
Team mein ek captain bhi choose karna hai
Total kitne tarike?
Pehle 5 choose karo — ¹⁰C5 = 252
Phir captain choose karo — 5 tarike
Total = 252 × 5 = 1260 tarike ✅
Real Life Mein Permutations aur Combinations
Roz ki zindagi mein bahut jagah use hota hai:
•Password — kitne possible passwords — Permutation.
•Lottery — jeetne ke chances — Combination.
•Cricket team — kitne tarike se team ban sakti hai — Combination.
•Menu — restaurant mein kitne meal combinations — Combination.
•Phone number — kitne possible numbers — Permutation.
Common Galtiyan — Jo Nahi Karni
Galti 1:
Permutation aur Combination mein confusion
Yaad karo — order matter karta hai — Permutation
Order matter nahi — Combination
Galti 2:
0! = 0 samajhna
❌ 0! = 0 — galat.
✅ 0! = 1 — hamesha.
Galti 3:
ⁿCr mein r! bhool jaana
Formula mein neeche r! × (n-r)! — dono.
Galti 4:
ⁿPr aur ⁿCr mein formula confuse karna
ⁿPr — neeche sirf (n-r)!
ⁿCr — neeche r! × (n-r)!
Yaad Rakhne Ka Aasan Tarika
Permutation vs Combination yaad karne ki trick:
•Permutation — Position matter karta hai — P se Position.
•Combination — Choose karo — order mat socho — C se Choose.
Jab bhi sawaal aaye — pehle socho:
“Kya order matter karta hai?”
Haan — Permutation.
Nahi — Combination.
Toh dosto, yeh tha Class 11 Permutations aur Combinations ka poora damdaar guide. Ek ek concept samjho — roz practice karo — board exam mein zaroor kaam aayega. Agar yeh article helpful laga to comment mein batao aur doston ke saath share karo.