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Binomial Theorem formulas

Master Binomial Theorem through 26 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Binomial Theorem, every formula

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Binomial expansion
$(x + a)^{n} = r = 0 \sum n^{n} C_{r} x^{n - r} a^{r}$
n positive integer
General term
$T_{r + 1} =^{n} C_{r} x^{n - r} a^{r}$
in (x+a)^n
Number of terms
$(x + a)^{n} has (n + 1) terms$
Expansion of (1+x)^n
$(1 + x)^{n} = r = 0 \sum n^{n} C_{r} x^{r}$
Middle term
$n even : (\frac{n}{2} + 1)^{th}; n odd : (\frac{n + 1}{2})^{th}, (\frac{n + 3}{2})^{th}$
General term of (x-a)^n
$T_{r + 1} = (- 1)^{r}^{n} C_{r} x^{n - r} a^{r}$
Symmetry of coefficients
$^{n} C_{r} =^{n} C_{n - r}$
Q1MCQSymmetry
${}^{10}C_3$ is equal to:
- A
  ${}^{10}C_7$
- B
  ${}^{10}C_4$
- C
  ${}^{10}C_6$
- D
  ${}^{10}C_2$
Sum of coefficients
$C_{0} + C_{1} + C_{2} + \dots + C_{n} = 2^{n}$
Alternating sum
$C_{0} - C_{1} + C_{2} - \dots + (- 1)^{n} C_{n} = 0$
Even and odd index sums
$C_{0} + C_{2} + C_{4} + \dots = C_{1} + C_{3} + \dots = 2^{n - 1}$
Independent term
$Independent of x at r = \frac{n p}{p + q} in (a x^{p} + \frac{b}{x ^{q}})^{n}$
Term containing x^m
$r = \frac{n p - m}{p + q} for (a x^{p} + \frac{b}{x ^{q}})^{n}$
Sum of squares of coefficients
$r = 0 \sum n C_{r}^{2} =^{2 n} C_{n}$
Weighted sum of coefficients
$r = 0 \sum n r C_{r} = n 2^{n - 1}$
Coefficient over (r+1)
$r = 0 \sum n \frac{C _{r}}{r + 1} = \frac{2 ^{n + 1} - 1}{n + 1}$
Q1MCQCoefficient over (r+1)
$\displaystyle\sum_{r=0}^{n}\dfrac{C_r}{r+1}$ equals:
- A
  $\dfrac{2^{\,n+1}-1}{n+1}$
- B
  $\dfrac{2^{n}-1}{n}$
- C
  $2^{n}$
- D
  $\dfrac{2^{n}}{n+1}$
Numerically greatest term
$Compute \frac{( n + 1 ) ∣ x ∣}{1 + ∣ x ∣} in (1 + x)^{n}$
Q1MCQNumerically greatest term
To find the numerically greatest term in $(1+x)^{n}$, one computes:
- A
  $\dfrac{(n+1)|x|}{1+|x|}$
- B
  $\dfrac{n|x|}{1+|x|}$
- C
  $\dfrac{(n+1)}{|x|}$
- D
  $n|x|$
Multinomial general term
$\frac{n !}{n _{1} ! n _{2} ! \dots n _{p} !} x_{1}^{n_{1}} x_{2}^{n_{2}} \dots x_{p}^{n_{p}}$
n_1+n_2+...+n_p = n
Q1MCQMultinomial term
The general term in $(x+y+z)^{n}$ is proportional to:
- A
  $\dfrac{n!}{a!\,b!\,c!}\,x^{a}y^{b}z^{c}$
- B
  ${}^{n}C_a\,x^{a}y^{b}z^{c}$
- C
  $\dfrac{n!}{a!}\,x^{a}$
- D
  $a!\,b!\,c!\,x^{a}y^{b}z^{c}$
Number of terms in a multinomial
$^{n + r - 1} C_{r - 1} terms in (a_{1} + a_{2} + \dots + a_{r})^{n}$
Series for rational index
$(1 + x)^{n} = 1 + n x + \frac{n ( n - 1 )}{2 !} x^{2} + \dots, ∣ x ∣ < 1$
n rational
Q1MCQRational-index series
For $|x|<1$, $(1+x)^{-1}$ equals:
- A
  $1-x+x^{2}-x^{3}+\cdots$
- B
  $1+x+x^{2}+\cdots$
- C
  $1-x^{2}+x^{4}-\cdots$
- D
  $1+2x+3x^{2}+\cdots$
Expansion of (1+x)^{-1}
$(1 + x)^{- 1} = 1 - x + x^{2} - x^{3} + \dots (∣ x ∣ < 1)$
Q1MCQCoefficient in (1+x)^{-1}
For $|x|<1$, the coefficient of $x^{r}$ in $(1+x)^{-1}$ is:
- A
  $(-1)^{r}$
- B
  $1$
- C
  $r+1$
- D
  $(-1)^{r}(r+1)$
Expansion of (1-x)^{-2}
$(1 - x)^{- 2} = 1 + 2 x + 3 x^{2} + \dots = r \geq 0 \sum (r + 1) x^{r}$
Approximation for small x
$(1 + x)^{n} \approx 1 + n x (x small)$
Divisibility
$(1 + α)^{n} - 1 is divisible by α$
Vandermonde coefficient
$coeff of x^{r} in (1 + x)^{m} (1 + x)^{n} =^{m + n} C_{r}$
Sum of all coefficients
$f (1) = sum of all coefficients of the polynomial$
Greatest binomial coefficient
$n even :^{n} C_{n /2}; n odd :^{n} C_{(n - 1) /2}$

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Binomial Theorem formulas

Binomial Theorem, every formula

Binomial expansion

General term

Number of terms

Expansion of (1+x)^n

Middle term

General term of (x-a)^n

Symmetry of coefficients

Sum of coefficients

Alternating sum

Even and odd index sums

Independent term

Term containing x^m

Sum of squares of coefficients

Weighted sum of coefficients

Coefficient over (r+1)

Numerically greatest term

Multinomial general term

Number of terms in a multinomial

Series for rational index

Expansion of (1+x)^{-1}

Expansion of (1-x)^{-2}

Approximation for small x

Divisibility

Vandermonde coefficient

Sum of all coefficients

Greatest binomial coefficient

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Other ways to revise this chapter

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