Permutations and Combinations
Bina ek-ek karke gine, "kitne tareekon se" kuch ho sakta hai — ye batana hi counting ka kaam hai. Permutations me arrangement (order) matter karta hai; combinations me sirf selection (order matter nahi karta).
1. Fundamental Principle of Counting
Agar ek kaam \(m\) tareekon se ho sakta hai, aur uske baad doosra kaam \(n\) tareekon se, to dono kaam mil ke \(m \times n\) tareekon se honge. (Yeh aur kaamon tak badhta hai.)
3 shirts aur 4 pants se kitne outfits ban sakte hain? \[ 3 \times 4 = 12 \text{ outfits}. \]
2. Factorial
\(n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1\), aur \(0! = 1\) (definition).
Jaise \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Useful: \(n! = n \times (n-1)!\).
3. Permutations (Order matter karta hai)
\(n\) alag objects me se \(r\) ko arrange karne ke tareeke: \[ {}^{n}P_{r} = \frac{n!}{(n-r)!}, \qquad 0 \le r \le n. \]
Khaas case: saare \(n\) objects arrange karne ke tareeke \(= {}^{n}P_{n} = n!\).
5 alag books me se 3 ko shelf par kitne tareekon se arrange karein? \[ {}^{5}P_{3} = \frac{5!}{(5-3)!} = \frac{120}{2} = 60. \]
Special cases
- Repetition allowed: \(n\) objects me se \(r\) ki arrangement \(= n^{r}\).
- Identical objects: \(n\) objects jisme \(p\) ek tarah ke, \(q\) doosre tarah ke ho, to distinct arrangements \(= \dfrac{n!}{p!\,q!}\).
Word "LEVEL" ke letters ki kitni distinct arrangements? Yahan L 2 baar, E 2 baar, V 1 baar (total 5). \[ \frac{5!}{2!\,2!} = \frac{120}{4} = 30. \]
4. Combinations (Order matter nahi karta)
\(n\) alag objects me se \(r\) ko choose karne ke tareeke: \[ {}^{n}C_{r} = \frac{n!}{r!\,(n-r)!}, \qquad 0 \le r \le n. \]
Permutation aur combination ka rishta:
\[ {}^{n}P_{r} = {}^{n}C_{r} \times r! \]
(Pehle \(r\) choose karo — \({}^{n}C_{r}\) tareeke — phir unhe arrange karo — \(r!\) tareeke.)
10 players me se 4 ki team kitne tareekon se chuni ja sakti hai? (Order matter nahi) \[ {}^{10}C_{4} = \frac{10!}{4!\,6!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210. \]
5. Useful Properties of \({}^{n}C_{r}\)
\[ {}^{n}C_{r} = {}^{n}C_{n-r} \]
\[ {}^{n}C_{r} + {}^{n}C_{r-1} = {}^{n+1}C_{r} \quad (\text{Pascal's rule}) \]
\[ {}^{n}C_{0} = {}^{n}C_{n} = 1 \]
Permutation vs Combination — kaise pehchaane?
- Agar arrangement / order / sequence / rank ki baat ho → Permutation.
- Agar sirf selection / group / team / choose ki baat ho → Combination.
Key Takeaways
- Multiplication principle counting ki base hai.
- \({}^{n}P_{r} = \dfrac{n!}{(n-r)!}\) (order matters), \({}^{n}C_{r} = \dfrac{n!}{r!(n-r)!}\) (order nahi).
- \({}^{n}P_{r} = {}^{n}C_{r} \cdot r!\).
- Identical objects me \(\dfrac{n!}{p!\,q!}\) use karo.
- \({}^{n}C_{r} = {}^{n}C_{n-r}\) aur Pascal's rule yaad rakho.