Binomial Theorem formulas for JEE Main & Advanced

23 formula entries for Binomial Theorem, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Binomial Theorem Statement

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

When to use. When expanding any binomial expression $(a + b)^{n}$ with a positive integer exponent.

Trap. Forgetting that the sum runs from $r = 0$ to $n$ , not $1$ to $n$ .

General Term in Binomial Expansion

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

When to use. To find any specific term in the expansion without writing all terms.

Trap. Confusing $T_{r + 1}$ with $T_{r}$ ; the index starts at $r = 0$ for the first term.

Finding Term Independent of x

For

(a x^{p} + b x^{- q})^{n}

T_{r + 1} = (\binom{n}{r}) (a x^{p})^{n - r} (b x^{- q})^{r}

, exponent of

x

p (n - r) - q r

. Set

p (n - r) - q r = 0

When to use. When a problem asks for the term that does not contain $x$ (constant term) in a binomial expansion.

Trap. Forgetting to consider the constant term from both factors; sometimes the exponent equation yields a non-integer $r$ (no independent term).

Middle Term(s) in Binomial Expansion

n

is even: middle term

T_{n / 2 + 1}

. If

n

is odd: middle terms

T_{(n + 1) / 2}

and

T_{(n + 3) / 2}

When to use. When asked to find the middle term(s) in the expansion of $(a + b)^{n}$ .

Trap. Using $n / 2$ as the term number instead of $n / 2 + 1$ .

Binomial Coefficient Properties

(\binom{n}{r}) = (\binom{n}{n - r})

;

\sum_{r = 0}^{n} (\binom{n}{r}) = 2^{n}

;

\sum_{r = 0}^{n} (- 1)^{r} (\binom{n}{r}) = 0

;

\sum_{r = 0}^{n} {(\binom{n}{r})}^{2} = (\binom{2 n}{n})

When to use. To simplify sums involving binomial coefficients, especially in combinatorial identities.

Trap. Confusing $\sum (\binom{n}{r}) = 2^{n}$ with $\sum {(\binom{n}{r})}^{2} = (\binom{2 n}{n})$ .

Greatest Term in Binomial Expansion

For

(1 + x)^{n}

\frac{T_{r + 1}}{T_{r}} = \frac{n - r + 1}{r} ∣ x ∣

. Solve

\frac{n - r + 1}{r} ∣ x ∣ \geq 1

to find

r

When to use. When asked to find the numerically greatest term in the expansion of $(1 + x)^{n}$ or $(a + b)^{n}$ .

Trap. Forgetting to consider absolute value of $x$ when $x$ is negative.

Binomial Theorem for Any Index

(1 + x)^{n} = \sum_{r = 0}^{\infty} \frac{n (n - 1) \dots (n - r + 1)}{r!} x^{r}

for

∣ x ∣ < 1

When to use. When expanding $(1 + x)^{n}$ where $n$ is not a positive integer, e.g., $\sqrt{1 + x}$ or $1 / (1 - x)$ .

Trap. Applying this for $∣ x ∣ \geq 1$ without convergence considerations.

Approximations Using Binomial Theorem

(1 + x)^{n} = 1 + n x + \frac{n (n - 1)}{2} x^{2} + O (x^{3})

When to use. When estimating values like $(1.01)^{5}$ or $(0.99)^{3}$ without a calculator.

Trap. Using the approximation when $x$ is not small (e.g., $x > 0.1$ leads to large error).

Finding Coefficient of a Specific Power

For

(a x^{p} + b x^{q})^{n}

, exponent of

x

T_{r + 1}

p (n - r) + q r

. Set equal to

k

and solve for

r

When to use. When asked for the coefficient of $x^{m}$ in a binomial expansion.

Trap. Forgetting to consider the coefficient from $a$ and $b$ as well.

Binomial Coefficient Notation

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

When to use. Whenever binomial coefficients appear in expansions or combinatorial problems.

Trap. Thinking $(\binom{n}{r}) = \frac{n!}{r!}$ (missing the $(n - r)!$ factor).

Sum of Coefficients in Binomial Expansion

\sum_{r = 0}^{n} (\binom{n}{r}) = 2^{n}

When to use. To quickly find the sum of all coefficients in any binomial expansion.

Trap. Forgetting that this works only for $(1 + x)^{n}$ ; for $(a + b x)^{n}$ , substitute $x = 1$ after factoring.

Sum of Coefficients of Even and Odd Terms

Sum of even terms =

\frac{(1 + 1)^{n} + (1 - 1)^{n}}{2} = 2^{n - 1}

; sum of odd terms =

\frac{(1 + 1)^{n} - (1 - 1)^{n}}{2} = 2^{n - 1}

When to use. When asked for the sum of coefficients of terms with even powers or odd powers.

Trap. Assuming the sum of even and odd terms are different; they are equal for $(1 + x)^{n}$ .

Using Binomial Theorem for Divisibility

Example:

6^{n} - 5 n - 1

is divisible by 25. Write

6 = 1 + 5

, so

6^{n} = 1 + 5 n + 25 (\dots)

When to use. When proving that a large number is divisible by another, e.g., $7^{2 n} - 1$ divisible by 48.

Trap. Not expanding fully to see the remainder pattern.

Multinomial Theorem Extension

(x_{1} + x_{2} + \dots + x_{m})^{n} = \sum \frac{n!}{r_{1}! r_{2}! \dots r_{m}!} x_{1}^{r_{1}} x_{2}^{r_{2}} \dots x_{m}^{r_{m}}

where

\sum r_{i} = n

When to use. When expanding expressions like $(a + b + c)^{n}$ or finding coefficients in such expansions.

Trap. Forgetting that the sum of exponents must equal $n$ .

Particular Term from the End

Term from end:

T_{n - r + 2} = (\binom{n}{n - r + 1}) a^{r - 1} b^{n - r + 1}

When to use. When asked for a term near the end of the expansion, e.g., 3rd term from the end.

Trap. Confusing $r$ -th from end with $r$ -th from beginning.

Binomial Theorem for Negative/Fractional Index

(1 + x)^{- n} = \sum_{r = 0}^{\infty} (- 1)^{r} (\binom{n + r - 1}{r}) x^{r}

When to use. When expanding $1 / (1 + x)^{n}$ or similar expressions.

Trap. Forgetting the alternating signs when $n$ is positive in the denominator.

Finding Numerically Greatest Term in $(a + b)^{n}$

Let

x = b / a

. Then

\frac{T_{r + 1}}{T_{r}} = \frac{n - r + 1}{r} ∣ x ∣

. Solve

\frac{n - r + 1}{r} ∣ x ∣ \geq 1

When to use. When asked for the numerically greatest term in $(a + b)^{n}$ .

Trap. Forgetting to take absolute value of $x$ .

Pascal's Triangle

(\binom{n}{r}) = (\binom{n - 1}{r - 1}) + (\binom{n - 1}{r})

When to use. To quickly generate binomial coefficients for small $n$ without factorials.

Trap. Thinking Pascal's triangle gives coefficients for any $n$ (only for positive integers).

Binomial Theorem in Series Summation

\sum_{r = 0}^{n} r (\binom{n}{r}) = n 2^{n - 1}

(from

x = 1

after differentiating

n (1 + x)^{n - 1}

When to use. When evaluating sums involving $r$ times binomial coefficients.

Trap. Forgetting to differentiate correctly; $\frac{d}{d x} (1 + x)^{n} = n (1 + x)^{n - 1}$ .

Binomial Theorem for Product of Two Binomials

Coefficient of

x^{m}

(1 + x)^{n} (1 + x)^{p} = \sum_{r = 0}^{m} (\binom{n}{r}) (\binom{p}{m - r})

When to use. When finding coefficients in the product of two binomial expansions.

Trap. Forgetting that the sum runs from $r = 0$ to $m$ , but $r$ cannot exceed $n$ and $m - r$ cannot exceed $p$ .

Binomial Theorem for Negative Exponents with Two Terms

(a + b x)^{- n} = a^{- n} \sum_{r = 0}^{\infty} (- 1)^{r} (\binom{n + r - 1}{r}) (b / a)^{r} x^{r}

When to use. When expanding expressions like $(2 - 3 x)^{- 4}$ .

Trap. Forgetting to factor out $a^{- n}$ correctly.

Finding Remainder Using Binomial Theorem