Binomial theorem — expansion, general term

WHAT is the binomial theorem?

There are exactly $==n+1==$ terms.
Powers of $a$ decrease $n \to 0$ ; powers of $b$ increase $0 \to n$ ; each term's exponents sum to $==n==$ .
The coefficients $\binom{n}{0},\binom{n}{1},\dots,\binom{n}{n}$ form the $n$ -th row of Pascal's triangle.

WHY does it look like this? (Derivation from scratch)

Write the product without collapsing anything: $(a+b)^n = \underbrace{(a+b)(a+b)\cdots(a+b)}_{n \text{ brackets}}$

To expand, we choose one letter from each bracket and multiply them. A generic term is a product of $n$ letters.

Question: how many of these products equal $a^{n-r}b^{r}$ ?

That happens when we pick $b$ from exactly $r$ brackets and $a$ from the other $n-r$ . The number of ways to choose which $r$ brackets give $b$ is $\binom{n}{r}$ .

Summing over all possible $r$ (from $0$ to $n$ ) gives the theorem. No formula was memorised — it fell out of counting. ∎

Figure — Binomial theorem — expansion, general term

HOW to use it: the general term

This single formula answers almost every exam question:

Find a specific term: plug in that $r$ .
Find the term containing $x^k$ : set up the exponent of $x$ equal to $k$ , solve for $r$ .
Find the term independent of $x$ (constant term): set the exponent of $x$ equal to $0$ .
Find the coefficient: compute $\binom{n}{r}\times$ (numeric part).

COMMON MISTAKES (steel-manned)

Imagine $n$ light switches, each either " $a$ -on" or " $b$ -on". Flipping all switches every possible way and writing down the result gives the expansion. To get "exactly $r$ switches on $b$ ", count how many ways to choose those $r$ switches — that's $\binom{n}{r}$ . Add up all the ways and you've built $(a+b)^n$ without memorising a thing.

Connections

Pascal's Triangle — the coefficients row by row.
Combinations nCr — the counting engine $\binom{n}{r}$ .
Factorials — how $\binom{n}{r}$ is computed.
Binomial Probability Distribution — same $\binom{n}{r}p^r q^{n-r}$ powers a random experiment.
Taylor & Maclaurin Series — generalised binomial $(1+x)^\alpha$ for non-integer $\alpha$ .
Permutations vs Combinations — why order doesn't matter here.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Binomial theorem ka core idea ekdum simple hai: jab tum $(a+b)$ ko $n$ baar multiply karte ho, to har term banane ke liye tumhe har bracket se ya to $a$ lena hai ya $b$ . Agar tum $b$ ko exactly $r$ brackets se choose karte ho, to woh choose karne ke tareeke $\binom{n}{r}$ hote hain — bas isi wajah se coefficient mein $\binom{n}{r}$ aata hai. Kuch ratna nahi, sirf counting samajhna hai.

Formula yaad rakho: $(a+b)^n=\sum \binom{n}{r}a^{n-r}b^r$ . Dhyaan do — $a$ ki power ghatti hai ( $n\to0$ ), $b$ ki power badhti hai ( $0\to n$ ), aur dono ka sum hamesha $n$ rehta hai. General term hai $T_{r+1}=\binom{n}{r}a^{n-r}b^r$ . Yeh $r+1$ isliye hai kyunki counter $r=0$ se start hota hai.

Exam mein sabse common questions: " $x^k$ wala term nikaalo" ya "constant term nikaalo". Trick same hai — pura term ek single power of $x$ ke roop mein likho, phir uski exponent ko $k$ (ya constant ke liye $0$ ) ke barabar rakh ke $r$ solve karo. Bas $r$ mil gaya to term ready.

Sabse badi galti: jab $a$ ya $b$ ke saath number ya power lagi ho (jaise $2x$ ya $\frac{-3}{x^2}$ ), tab coefficient sirf $\binom{n}{r}$ nahi hota — usmein $2^{n-r}$ , $(-3)^r$ jaise numeric factors bhi multiply hote hain. Aur negative sign ko kabhi mat bhoolna, warna answer ka sign ulta ho jaayega. Yeh dhyaan rakhoge to full marks pakke.

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

Binomial theorem — expansion, general term

WHAT is the binomial theorem?

WHY does it look like this? (Derivation from scratch)

HOW to use it: the general term

WORKED EXAMPLES

COMMON MISTAKES (steel-manned)

Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Statistics & Probability — Intermediate

Connections