Binomial theorem — expansion, general term

2.7.12 · Maths › Statistics & Probability — Intermediate

Binomial theorem HAI kya?

Exactly $== n + 1 ==$ terms hote hain.
$a$ ki powers ghatti hain $n \to 0$ ; $b$ ki powers badhti hain $0 \to n$ ; har term ke exponents ka sum $== n ==$ hota hai.
Coefficients $(0 n), (1 n), \dots, (n n)$ Pascal's triangle ki $n$ -vi row banate hain.

Yeh aisa kyun dikhta hai? (Scratch se Derivation)

Product ko collapse kiye bina likho: $(a + b)^{n} = n brackets (a + b) (a + b) \dots (a + b)$

Expand karne ke liye, har bracket se ek letter choose karo aur unhe multiply karo. Ek generic term $n$ letters ka product hota hai.

Sawaal: in products mein se kitne $a^{n - r} b^{r}$ ke barabar hain?

Yeh tab hota hai jab hum exactly $r$ brackets se $b$ choose karein aur baaki $n - r$ se $a$ . Yeh choose karne ke ki kaun se $r$ brackets $b$ denge, tarike hain $(r n)$ .

Sab possible $r$ ( $0$ se $n$ tak) par sum karne se theorem milti hai. Koi formula yaad nahi kiya — yeh counting se hi nikal aaya. ∎

Figure — Binomial theorem — expansion, general term

Ise USE kaise karein: general term

Yeh ek formula almost har exam question ka jawab deta hai:

Koi specific term dhundho: woh $r$ plug in karo.
$x^{k}$ wala term dhundho: $x$ ka exponent $k$ ke barabar set karo, $r$ solve karo.
$x$ se independent term dhundho (constant term): $x$ ka exponent $0$ ke barabar set karo.
Coefficient dhundho: $(r n) \times$ (numeric part) calculate karo.

COMMON MISTAKES (steel-manned)

Active recall

Recall Reveal karne se pehle predict karo (Forecast-then-Verify)

$(a + b)^{12}$ mein kitne terms hain? → $13$ .
$(x - \frac{1}{x})^{10}$ mein, kaun sa $r$ constant term deta hai? Set karo $10 - 2 r = 0 \Rightarrow r = 5$ .
$(1 + 2 x)^{5}$ mein $x^{3}$ ka coefficient? $(3 5) 2^{3} = 10 \cdot 8 = 80$ .

(a + b)^{n}

ki binomial expansion kya hai?

\sum_{r = 0}^{n} (r n) a^{n - r} b^{r}

(r n)

coefficient ki tarah kyun aata hai?

Yeh un brackets ki sankhya count karta hai jisme se hum

b

choose karte hain (baaki

a

dete hain), jo sab same

a^{n - r} b^{r}

term produce karte hain.

(a + b)^{n}

mein kitne terms hote hain?

n + 1

(a + b)^{n}

ka general term likho.

T_{r + 1} = (r n) a^{n - r} b^{r}

r = 0, 1, \dots, n

T_{r + 1}

kyun hai,

T_{r}

kyun nahi?

Kyunki counter

r

0

se shuru hota hai;

T_{1}

hai

r = 0

, toh subscript ek aage hota hai.

x^{k}

wala term kaise dhundho?

T_{r + 1}

x

ki single power ke roop mein likho, uska exponent

= k

set karo,

r

solve karo.

x

se independent term kaise dhundho?

T_{r + 1}

mein

x

ka exponent

0

ke barabar set karo aur

r

solve karo.

(1 + x)^{n}

mein

x^{r}

ka coefficient?

(r n)

(1 + x)^{n}

ka middle term jab

n

even ho?

Single term

T_{n /2 + 1} = (n /2 n) x^{n /2}

(2 x - 3)^{n}

mein, kya coefficient sirf

(r n)

hai?

Nahi — numeric factors include karo:

(r n) (2)^{n - r} (- 3)^{r}

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho $n$ light switches hain, har ek ya toh " $a$ -on" ya " $b$ -on" hai. Sab switches ko har possible tarike se flip karke result likhne se expansion milti hai. "Exactly $r$ switches $b$ par" paane ke liye, count karo ki woh $r$ switches choose karne ke kitne tarike hain — woh hai $(r n)$ . Sab tarike add karo aur tumne $(a + b)^{n}$ bana liya bina kuch yaad kiye.

Connections

Pascal's Triangle — coefficients row by row.
Combinations nCr — counting engine $(r n)$ .
Factorials — $(r n)$ kaise compute hota hai.
Binomial Probability Distribution — same $(r n) p^{r} q^{n - r}$ ek random experiment ko power deta hai.
Taylor & Maclaurin Series — generalised binomial $(1 + x)^{α}$ non-integer $α$ ke liye.
Permutations vs Combinations — yahaan order kyun matter nahi karta.

$r$	$(r 4)$	term
0	1	$x^{4}$
1	4	$4 x^{3} (2) = 8 x^{3}$
2	6	$6 x^{2} (4) = 24 x^{2}$
3	4	$4 x (8) = 32 x$
4	1	$16$