Permutations — nPr, arrangements with restrictions: Foundations

2.7.10 · D1 · Maths › Statistics & Probability — Intermediate › Permutations — nPr, arrangements with restrictions

Ye page topic Permutations — nPr, arrangements with restrictions (ise neeche the topic kahenge) ke liye ek self-contained foundation hai. Wo topic poochhhta hai "kitne ordered arrangements ban sakte hain?" aur kuch symbols aur ideas use karta hai bina ruke unhe build kiye. Yahan hum un sab ko zero se build karte hain, ek aisi order mein jahan har piece agla piece earn karta hai. Upar se neeche padho; kuch bhi skip mat karo — koi bhi symbol define hone se pehle use nahi hota.

0. "Object" aur "arrangement" — raw picture

Paanch doston aur zameen par paanch chalk marks ko imagine karo jinhein slot 1, slot 2, slot 3, slot 4, slot 5 label kiya gaya hai. Ek arrangement ek poora tarika hai ek ek dost ko har mark par rakhne ka.

Figure s01 — Left side par paanch distinct labelled objects ka ek pile (woh cheezein jo hum arrange karenge). Right side par paanch empty ordered slots. Ek arrow dikhata hai ek object pile se nikal kar ek slot mein ja raha hai: ek arrangement ek poora tarika hai har slot fill karne ka, ek ek object se.

Topic ko yeh kyun chahiye: poora subject hai "kitni alag alag aisi pictures exist karti hain?" Agar tum slots nahi dekh sako, koi bhi symbol kuch matlab nahi rakhega.

1. Woh letter jo tumhare objects count karta hai

Figure s01 mein left-hand pile dekho — $n$ exactly unhi objects ko count karta hai, koi bhi place hone se pehle.

Number ki jagah letter kyun? Kyunki ek formula kisi bhi pile size ke liye kaam karna chahiye. $n$ likhne se ek hi rule 5 letters, 10 logon, ya ek million cards ko cover karta hai.

2. Woh letter jo tumhare slots count karta hai

Figure s01 mein, agar sirf pehle teen chalk marks use kiye jaate aur aakhiri do empty rehte, toh woh $n = 5$ objects mein se $r = 3$ slots fill hone ka case hota.

3. Curly braces ${}$ aur "distinct" word

"Distinct" yahan kyun matter karta hai: topic ka formula $n P_{r} = \frac{n !}{( n - r )!}$ silently assume karta hai koi duplicates nahi hain. Agar do objects identical hote, unhe swap karna ek naya picture nahi banata, aur tum over-count karte. (Woh repaired case hai Permutations with Repetition.)

4. Multiplication principle — the engine

Yahi poora Fundamental Counting Principle hai, aur isi wajah se hum choices ko add karne ki bajay multiply karte hain.

Figure s02 — "2 shirts, 3 hats" ke liye ek branching tree. Start dot se, do magenta branches (choice 1) do shirts tak pahunchti hain; har shirt se, teen orange branches (choice 2) teen hats tak pahunchti hain. Endpoints count karne se $2 \times 3 = 6$ complete paths milte hain — har path ek poora outfit hai.

Multiply kyun karein, add kyun nahi? Figure s02 mein tree dekho. Adding jawaab deta "kitne single decisions exist karte hain." Multiplying jawaab deta "kitne complete paths upar se neeche tak exist karte hain" — aur ek poori arrangement ek complete path hai. Yahi hum count karna chahte hain.

5. Factorial symbol $!$

Topic ko $!$ kyun chahiye: agar tum sab $n$ slots fill karo, toh choices hain $n, n - 1, n - 2, \dots, 1$ . Unhe multiply karna (multiplication principle!) literally $n!$ ki definition hai. Toh ek factorial bas hai "sabko line mein khada karne ke tareekon ki count."

Special value $0!$

Yeh koi trick nahi hai; yeh woh ek hi value hai jo formulas ko consistent rakhti hai (dekho Factorials and 0!). Isse $n P_{n} = n!$ aur $n P_{0} = 1$ sahi nikalta hai.

6. Formula ke andar subtraction: $n - r$ aur $n - r + 1$

Figure s03 — Usi pile ke teen snapshots left to right. Slot 1 par $n$ objects aur $n$ choices hain; slot 2 par pile ne ek khoya hai, $n - 1$ dikhta hai; slot 3 par $n - 2$ dikhta hai. Snapshots ke beech arrows mark karte hain "ek placement ek object remove karti hai", isliye choices count down hoti hain.

$(n - r)!$ compact formula mein kyun aata hai: jab tum $r$ objects place kar chuke ho, pile mein $n - r$ abhi bhi hain, unused. Factorial $(n - r)!$ un leftover objects ke baaki bachne wale tareekon ko count karta hai jis mein woh arrange ho sakte the — aur usse divide karna exactly un arrangements ko remove karta hai jinhe tum count karna nahi chahte. Yahi ek algebra move hai jo lambe product ko $\frac{n !}{( n - r )!}$ mein badalta hai.

7. Notation $n P_{r}$ khud

$n P_{r} = r factors n (n - 1) \dots (n - r + 1) = \frac{n !}{( n - r )!}$

Sirf ek number kyun nahi kehte? Kyunki sawaal $n$ aur $r$ ke saath badalta hai. Symbol poori slot-filling story ko teen characters mein pack karta hai.

8. Order vs no-order — raaste ka fork

$A B$ aur $B A$ socho: woh permutation list par do alag arrangements hain, lekin combination list par ek selection mein collapse ho jaate hain. Unke beech link hai $n C_{r} = \frac{n P _{r}}{r !}$ — tum un $r!$ internal orderings ko divide out karte ho jinhe ab tum care nahi karte. Yeh exactly §6 se leftover-factorial idea hai, reuse hua. (Details Combinations — nCr mein.)

Prerequisite map

Neeche diagram dikhata hai ki yeh foundations topic ko kaise feed karte hain. Har arrow padho as "left box pehle chahiye right box sense banane se pehle." Words mein: objects aur slots (§0) counters $n$ (§1) aur $r$ (§2) ko janam dete hain; woh dono multiplication principle (§4) ko feed karte hain; multiplication principle shrinking pile (§6) produce karta hai, jo turn mein factorials aur $0!$ (§5) define karta hai; factorials aur multiplication principle milkar $n P_{r}$ (§7) assemble karte hain; aur order-matters question (§8) woh gate hai jo decide karta hai ki tum $n P_{r}$ kar rahe ho. Agar picture load na ho, woh sentence poora map hai.

Topic $n P_{r}$ in sab prerequisites ke munh par baitha hai.

Equipment checklist

Self-test: kya tum reveal karne se pehle har jawaab de sakte ho? Agar koi bhi shaky lage, woh section dobara padho.

n

kya stand karta hai?

Available distinct objects ki total count (pile ka size).

r

kya stand karta hai?

Un positions ki count jo tum actually fill karte ho;

r \leq n

satisfy karna zaroori hai.

Hum choices ko add karne ki bajay multiply kyun karte hain?

Har poori arrangement decisions ka ek complete path hai; multiply karna sab paths count karta hai, add karna sirf single decisions count karta hai.

k!

ka matlab kya hai?

k \times (k - 1) \times \dots \times 2 \times 1

—

k

distinct objects ko line mein lagaane ke tareekon ki count.

0!

kya hai aur kyun?

0! = 1

; kuch arrange karne ka exactly ek tarika hai (empty arrangement), aur yeh

n P_{n} = n!

ko consistent rakhta hai.

r

-ve slot par pile

n - r + 1

tak kyun shrink hoti hai?

Har placed object pile se chala jaata hai, isliye

r - 1

placements ke baad sirf

n - (r - 1) = n - r + 1

bachi rehti hai.

\frac{n !}{( n - r )!}

mein

(n - r)!

kyun aata hai?

Yeh

n - r

unused leftover objects ke arrangements count karta hai, jinhe hum divide out karte hain kyunki humne unhe kabhi place nahi kiya.

n P_{r}

kaun sa sawaal answer karta hai?

n

distinct objects mein se

r

objects ke kitne ordered arrangements ban sakte hain, bina repetition ke.

5 P_{3}

ko successive choices se nikalo.

5 \times 4 \times 3 = 60

, jo

\frac{5 !}{2 !} = 60

se match karta hai.

Permutations aur combinations ko alag karne wala ek sawaal?

"Kya do chosen items ko swap karna ek naya outcome banata hai?" Haan → permutation; Nahi → combination.

"Distinct objects" hume kya assume karne deta hai?

Koi bhi do objects identical nahi hain, isliye har swap genuinely naya arrangement banata hai (koi over-counting nahi).

Connections

Fundamental Counting Principle — §4 ka multiplication engine.
Factorials and 0! — symbol $!$ aur kyun $0! = 1$ .
Combinations — nCr — §8 se "order ignored" fork.
Permutations with Repetition — kya hota hai jab objects distinct nahi hote.
Circular Permutations — line ki bajay ring mein arrangements.
Probability — Equally Likely Outcomes — jahan yeh counts denominators ban jaate hain.
Permutations — nPr, arrangements with restrictions — woh parent jiske liye yeh page tumhe taiyaar karta hai.