Combinations — nCr, Pascal's triangle: Foundations

2.7.11 · D1 · Maths › Statistics & Probability — Intermediate › Combinations — nCr, Pascal's triangle

Yeh page assume karti hai ki aap kuch nahi jaante aur har woh symbol build karti hai jo parent note use karta hai, ek brick at a time. Upar se neeche padhein: har idea ek floor hai jis par agla khada hota hai.

1. Distinct cheezein count karna — pehli picture

Kisi bhi formula se pehle, humein distinct item ka idea chahiye: ek cheez jise aap har doosri cheez se alag pehchaan sako. Teen dost jinka naam Amit, Bina, Chetan hai woh distinct hain — aap unme se har ek ki taraf point kar sakte hain.

Yeh topic ko kyun chahiye: parent formula $n$ distinct items ke subsets count karta hai. Agar items repeat ho sakti hain ya identical hoti, toh "kitne tarike se choose karein" ka matlab alag hota. Isliye "distinct" sabse pehla floor hai.

2. Symbol $n$ — "mere paas kitni hain"

Hum fixed number ki jagah letter isliye use karte hain taaki ek formula ek saath kisi bhi size ke pile ke baare mein baat kar sake. Yahi poora reason hai ki algebra mein letters use hote hain: ek baar likhein, hamesha ke liye use karein.

3. Symbol $r$ — "main kitne leta/leti hoon"

Dono letters kyun? Parent note baar baar $(r n)$ jaise cheezein likhta hai — " $n$ mein se, $r$ choose karo." Aapko in dono roles ko feel karna chahiye: $n$ pool hai, $r$ grab hai.

4. Order matters vs. order doesn't — sab kuch ka dil

Yahi woh ek distinction hai jis par yeh poora topic ghoomta hai, isliye picture dheeray se dekho.

Yeh topic ko kyun chahiye: ek permutation ( $^{n} P_{r}$ ) ordered count hai; ek combination ( $(r n)$ ) unordered count hai. Parent page ka central move — $r!$ se divide karna — sirf isliye exist karta hai taaki pehle ko doosre mein badla ja sake.

Recall Curly braces

${}$ Jab hum curly braces ke saath ek group likhte hain ${a, b}$ , toh braces ek signal hain: "mere andar order irrelevant hai." Isliye ${a, b}$ aur ${b, a}$ literally same set hain.

5. Exclamation mark: factorial $n!$

Iske peeche ki picture — factorial arrangements count karta hai. 3 distinct logo ko line up karne ke kitne tarike hain? Pehle slot mein 3 choices hain, agle mein 2 bache, aakhri mein 1 bacha: $3 \times 2 \times 1 = 3! = 6.$

Yeh topic ko kyun chahiye: $r$ items ke ek group ke orderings ki number exactly $r!$ hoti hai. Yahi woh "un-shuffle" number hai jisse aap divide karte hain. Poore engine ke liye Factorials and 0! dekho.

6. Falling products — $n (n - 1) (n - 2) \dots$

Parent note likhta hai $^{n} P_{r} = n (n - 1) (n - 2) \dots (n - r + 1)$ , " $r$ factors." Aaiye ise samjhein.

Picture — $r$ slots ko shrinking pool se bharna: slot 1 mein $n$ choices hain, slot 2 mein $n - 1$ (ek use ho gaya), aur aise $r$ slots tak. Woh product hi $^{n} P_{r}$ hai.

7. Choose symbol $(r n)$

Picture: yeh pockets ki count hai, line-ups ki nahi. Jab bhi answer ko tab nahi badalna chahiye jab aap chosen items ko shuffle karo, yahi aapka symbol hai.

8. Pascal's triangle notation ki sigma-free reading

Parent numbers ko "row $n$ , position $r$ " ki tarah draw karta hai. Do conventions jo aapko pakka karne chahiye:

Zero-indexing kyun? Kyunki $(0 n) = 1$ har row ki pehli entry honi chahiye, aur $(0 0) = 1$ akela top entry hai. 0 se count shuru karna triangle ke edges ko hamesha $1$ banata hai, jo "kuch nahi choose karo / sab kuch choose karo" se match karta hai. Dekho Pascal's Triangle patterns.

9. Woh plus sign jo triangle ko stack karta hai

Pascal ka rule $(r n) = (r - 1 n - 1) + (r n - 1)$ ordinary addition use karta hai, lekin reason kyu aap add karte hain woh counting law hai "disjoint cases add up."

Yahi "cases add, complement subtracts" idea parent ke "at least one" example aur Probability — counting outcomes ke saare cases mein kaam karta hai.

Prerequisite map

Ise ek staircase ki tarah padhein: distinctness aur dono letters $n, r$ pehle aate hain; factorials (aur $0!$ ) permutations ko power karte hain; permutations plus "order doesn't matter" combinations dete hain; combinations row wise lagaye jaate hain toh Pascal's triangle milta hai; aur "cases add" idea Pascal's rule deta hai.

Equipment checklist

Khud test karo — right side cover karo aur reveal karne se pehle jawab do.

"Distinct items" ka matlab kya hai?

Objects jinhe aap alag pehchaan sako; unke names swap karne se badal jaata hai ki kaun sa aap mean kar rahe hain.

(r n)

mein

n

kya hai aur

r

kya hai?

n

= pool ki size jisme se aap choose karte hain;

r

= aap kitne lete hain, with

0 \leq r \leq n

Permutation aur combination mein ek kya fark hai?

Permutation ordered line-ups count karta hai; combination unordered groups count karta hai (order irrelevant).

4!

calculate karo.

4 \cdot 3 \cdot 2 \cdot 1 = 24

0! = 1

kyun hai?

Kuch bhi arrange karne ka exactly ek tarika hota hai — khaali arrangement.

^{7} P_{3}

ka falling-product form aur uski value likhein.

7 \cdot 6 \cdot 5 = 210

(teen factors, aakhri hai

7 - 3 + 1 = 5

^{n} P_{r}

(r n)

banao.

(r n) = \frac{^{n} P _{r}}{r !} = \frac{n !}{r ! ( n - r )!}

Pascal's triangle mein entry

(2 5)

ki row aur position kya hain, aur uski value?

Row

n = 5

, position

r = 2

; value

10

Do Pascal-rule terms add kyun hote hain, multiply kyun nahi?

Woh disjoint, exhaustive cases hain (item IN vs OUT), aur disjoint cases add hote hain.

{a, b} = {b, a}

aapko kya bataata hai?

Curly braces matlab andar ka order irrelevant hai — yeh same set hai.

Connections

Permutations — nPr — ordered count jise combinations divide karta hai.
Factorials and 0! — multiplication engine aur kyun $0! = 1$ .
Binomial Theorem — jahan yeh $(r n)$ coefficients ke roop mein dikhte hain.
Probability — counting outcomes — "cases add, complement subtracts" action mein.
Pascal's Triangle patterns — row/position grid kya reveal karta hai.
Combinations — nCr, Pascal's triangle — parent topic jiske liye yeh page aapko prepare karta hai.