Binomial Theorem Formula Sheet

Every must-know Binomial Theorem result in one scannable place — expansion, general and middle term, coefficient identities, the multinomial theorem, and approximation formulas. Use this for last-minute JEE Main and Advanced revision.

The Binomial Theorem

For any positive integer $n$:

$$\boxed{(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r}$$$$= \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + \binom{n}{n}b^n$$

where the binomial coefficient is

$$\binom{n}{r} = {}^nC_r = \frac{n!}{r!(n-r)!}$$

Standard forms

Form	Expansion
$(1 + x)^n$	$1 + nx + \dfrac{n(n-1)}{2!}x^2 + \dfrac{n(n-1)(n-2)}{3!}x^3 + \cdots + x^n$
$(1 - x)^n$	$1 - nx + \dfrac{n(n-1)}{2!}x^2 - \cdots + (-1)^n x^n$
$(x + a)^n$	$x^n + nx^{n-1}a + \dfrac{n(n-1)}{2!}x^{n-2}a^2 + \cdots + a^n$

General Term & Term Finding

The general term — the $(r+1)^{\text{th}}$ term — of $(a+b)^n$:

$$\boxed{T_{r+1} = \binom{n}{r} a^{n-r} b^r}, \quad r = 0, 1, 2, \ldots, n$$

Expansion	General term $T_{r+1}$
$(x + a)^n$	$\binom{n}{r} x^{n-r} a^r$
$(1 + x)^n$	$\binom{n}{r} x^r$
$(ax + b)^n$	$\binom{n}{r} a^{n-r} b^r x^{n-r}$
$\left(x + \tfrac{1}{x}\right)^n$	$\binom{n}{r} x^{n-2r}$

Term from the end

$$r^{\text{th}} \text{ term from the end} = (n - r + 2)^{\text{th}} \text{ term from the start}$$

Middle Term(s)

Case	Middle term(s)
$n$ even	one term: $T_{\frac{n}{2}+1}$
$n$ odd	two terms: $T_{\frac{n+1}{2}}$ and $T_{\frac{n+3}{2}}$

For $n$ even:

$$\boxed{T_{\frac{n}{2}+1} = \binom{n}{n/2}\, a^{n/2} b^{n/2}}$$

For $n$ odd:

$$\boxed{T_{\frac{n+1}{2}} = \binom{n}{(n-1)/2}\, a^{(n+1)/2} b^{(n-1)/2}}$$$$\boxed{T_{\frac{n+3}{2}} = \binom{n}{(n+1)/2}\, a^{(n-1)/2} b^{(n+1)/2}}$$

Greatest coefficient: $\binom{n}{n/2}$ if $n$ is even; $\binom{n}{(n-1)/2} = \binom{n}{(n+1)/2}$ if $n$ is odd.

Greatest Term (Numerically)

For $(a + b)^n$, the numerically greatest term is $T_{r+1}$ where

$$\boxed{r = \left[\frac{(n+1)\,|b|}{|a| + |b|}\right]} \quad ([\,\cdot\,] = \text{greatest integer})$$

Equivalently, $T_{r+1} > T_r$ as long as

$$\frac{(n - r + 1)\,|b|}{r\,|a|} > 1$$

For $(1+x)^n$ with $x > 0$: if $\dfrac{(n+1)x}{1+x}$ is an integer $m$, then $T_m$ and $T_{m+1}$ are both greatest; otherwise $T_{[m]+1}$ is greatest.

Binomial Coefficient Properties

$$\binom{n}{0} = \binom{n}{n} = 1, \quad \binom{n}{1} = \binom{n}{n-1} = n$$

Identity	Formula
Symmetry	$\binom{n}{r} = \binom{n}{n-r}$
Pascal’s identity	$\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$
Sum of consecutive	$\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1}$
Ratio of consecutive	$\dfrac{\binom{n}{r}}{\binom{n}{r-1}} = \dfrac{n-r+1}{r}$
Index relation	$r\binom{n}{r} = n\binom{n-1}{r-1}$

Sum Identities (with $C_r = \binom{n}{r}$)

Sum	Result
$C_0 + C_1 + C_2 + \cdots + C_n$	$2^n$
$C_0 - C_1 + C_2 - \cdots + (-1)^n C_n$	$0$
$C_0 + C_2 + C_4 + \cdots$	$2^{n-1}$
$C_1 + C_3 + C_5 + \cdots$	$2^{n-1}$
$C_1 + 2C_2 + 3C_3 + \cdots + nC_n = \sum r\,C_r$	$n\cdot 2^{n-1}$
$\sum_{r=0}^{n} r^2 \binom{n}{r}$	$n(n+1)\,2^{n-2}$
$C_0^2 + C_1^2 + \cdots + C_n^2 = \sum \binom{n}{r}^2$	$\binom{2n}{n}$

Headline results:

$$\boxed{\sum_{r=0}^{n}\binom{n}{r} = 2^n} \qquad \boxed{\sum_{r=0}^{n}\binom{n}{r}^2 = \binom{2n}{n}}$$$$\boxed{\sum_{r=0}^{n} r\binom{n}{r} = n\cdot 2^{n-1}}$$

Vandermonde’s identity

$$\boxed{\sum_{r=0}^{k} \binom{m}{r}\binom{n}{k-r} = \binom{m+n}{k}}$$

Integration / differentiation results

$$\sum_{r=0}^{n} \frac{\binom{n}{r}}{r+1} = \frac{2^{n+1} - 1}{n+1}$$$$\sum_{r=0}^{n}(-1)^r\binom{n}{r}^2 = \begin{cases} 0 & n \text{ odd} \\ (-1)^{n/2}\binom{n}{n/2} & n \text{ even} \end{cases}$$

Multinomial Theorem

For a positive integer $n$ and $k$ variables:

$$\boxed{(x_1 + x_2 + \cdots + x_k)^n = \sum \frac{n!}{n_1!\, n_2! \cdots n_k!}\, x_1^{n_1} x_2^{n_2} \cdots x_k^{n_k}}$$

where $n_1 + n_2 + \cdots + n_k = n$, all $n_i \ge 0$.

Multinomial coefficient:

$$\binom{n}{n_1, n_2, \ldots, n_k} = \frac{n!}{n_1!\, n_2! \cdots n_k!} = \binom{n}{n_1}\binom{n-n_1}{n_2}\binom{n-n_1-n_2}{n_3}\cdots$$

Trinomial special case:

$$(a + b + c)^n = \sum_{p+q+r=n} \frac{n!}{p!\,q!\,r!}\, a^p b^q c^r$$

Quantity	Formula	Notes
Number of terms	$\binom{n+k-1}{k-1} = \binom{n+k-1}{n}$	stars-and-bars; $\binom{n+2}{2}$ for trinomial
Sum of all coefficients	$k^n$	put every variable $= 1$
Greatest coefficient	powers as equal as possible	if $n = kq + r$: $r$ terms get power $q+1$, rest get $q$

Approximations (small $x$)

For $|x| < 1$ and any real $n$ (positive, negative, or fractional):

$$\boxed{(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots}$$

• First order: $(1+x)^n \approx 1 + nx$
• Second order: $(1+x)^n \approx 1 + nx + \dfrac{n(n-1)}{2}x^2$

Expression	Approximation (small $x$)
$(1 + x)^n$	$1 + nx$
$(1 - x)^n$	$1 - nx$
$\sqrt{1 + x}$	$1 + \dfrac{x}{2}$
$\sqrt[3]{1 + x}$	$1 + \dfrac{x}{3}$
$\dfrac{1}{1 + x}$	$1 - x$
$\dfrac{1}{\sqrt{1 + x}}$	$1 - \dfrac{x}{2}$
$\dfrac{1}{(1 + x)^2}$	$1 - 2x$

Higher-order single-variable expansions used in JEE:

$$\sqrt{1+x} \approx 1 + \frac{x}{2} - \frac{x^2}{8}, \qquad \sqrt[3]{1+x} \approx 1 + \frac{x}{3} - \frac{x^2}{9}$$$$\frac{1}{1+x} = (1+x)^{-1} \approx 1 - x + x^2 - x^3 + \cdots$$

Relative error when using $(1+x)^n \approx 1 + nx$:

$$\text{Relative error} \approx \frac{n(n-1)x^2}{2}$$

Pascal’s Triangle

Each entry is the sum of the two directly above it (Pascal’s identity), giving the binomial coefficients row by row.

graph TD
    A["n=0:  1"] --> B["n=1:  1  1"]
    B --> C["n=2:  1  2  1"]
    C --> D["n=3:  1  3  3  1"]
    D --> E["n=4:  1  4  6  4  1"]
    E --> F["n=5:  1  5  10  10  5  1"]

One-Glance Summary

Quantity	Formula
Expansion	$(a+b)^n = \sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r$
Number of terms	$n + 1$
General term	$T_{r+1} = \binom{n}{r}a^{n-r}b^r$
Middle term ($n$ even)	$T_{\frac{n}{2}+1}$
Middle terms ($n$ odd)	$T_{\frac{n+1}{2}},\ T_{\frac{n+3}{2}}$
Greatest term index	$r = \left[\dfrac{(n+1)
Symmetry	$\binom{n}{r} = \binom{n}{n-r}$
Pascal’s identity	$\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$
Ratio	$\dfrac{\binom{n}{r}}{\binom{n}{r-1}} = \dfrac{n-r+1}{r}$
$\sum C_r$	$2^n$
$\sum (-1)^r C_r$	$0$
Even = odd place sum	$2^{n-1}$
$\sum r\,C_r$	$n\,2^{n-1}$
$\sum r^2 C_r$	$n(n+1)2^{n-2}$
$\sum C_r^2$	$\binom{2n}{n}$
Vandermonde	$\sum_r \binom{m}{r}\binom{n}{k-r} = \binom{m+n}{k}$
Multinomial terms	$\binom{n+k-1}{k-1}$
Multinomial coeff. sum	$k^n$
Approximation	$(1+x)^n \approx 1 + nx,\

Related Topics

• Binomial Theorem for Positive Integer
• General Term and Middle Term
• Properties of Binomial Coefficients
• Multinomial Theorem
• Applications in Approximations
• Permutations and Combinations
• Sequences and Series