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Binomial Theorem and Its Simple Applications Formula Sheet for JEE

33+ JEE formulas in this unit

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The Binomial Theorem and Its Simple Applications JEE formula sheet lists 33+ important formulas for JEE Main and Advanced, including essential identities from Binomial theorem for a positive integral index, General term and middle term and simple applications. Revise essential formulas first, then practise MCQs on Goodmarks.

Download-free JEE mathematics formula revision for Binomial Theorem and Its Simple Applications. This unit-wise formula list covers 33+ exam-relevant results across Binomial theorem for a positive integral index, General term and middle term and simple applications, organised by subtopic for quick last-minute revision.

JEE Formula Sheet

33 formulas across 2 subtopics — organised for JEE Main & Advanced revision

Practise MCQs for this unit
Essential: 15Important: 13Supplementary: 5

Binomial theorem for a positive integral index

Binomial theoremEssential
(x+a)n=∑r=0n(nr)xn−rar=(n0)xn+(n1)xn−1a+⋯+(nn)an(x+a)^n=\sum_{r=0}^{n}{n\choose r}x^{n-r}a^r={n\choose 0}x^n+{n\choose 1}x^{n-1}a+\cdots+{n\choose n}a^n
Variables
nn is a positive integer; x,ax,a are any numbers/expressions.
Conditions
Applicable for positive integral nn.
Where used in JEE
Expansion of powers, coefficient finding, term identification, simplification.
Binomial coefficient formulaEssential
(nr)=n!r!(n−r)!{n\choose r}=\frac{n!}{r!(n-r)!}
Variables
n∈Nn\in\mathbb{N}, rr is an integer with 0≤r≤n0\le r\le n.
Conditions
Defined for integers rr in the stated range.
Where used in JEE
Computing coefficients in binomial expansions.
Symmetry of binomial coefficientsEssential
(nr)=(nn−r){n\choose r}={n\choose n-r}
Variables
nn is a positive integer; 0≤r≤n0\le r\le n.
Where used in JEE
Simplifying coefficients, identifying equal coefficients, middle term analysis.
Pascal recurrenceImportant
(nr)+(nr−1)=(n+1r){n\choose r}+{n\choose r-1}={n+1\choose r}
Variables
nn is a non-negative integer; 1≤r≤n1\le r\le n.
Where used in JEE
Coefficient manipulation, proving identities, recursive computation.
Boundary values of binomial coefficientsImportant
(n0)=(nn)=1,(n1)=(nn−1)=n{n\choose 0}={n\choose n}=1,\qquad {n\choose 1}={n\choose n-1}=n
Variables
nn is a positive integer.
Where used in JEE
Writing edge terms quickly in expansions.
Consecutive coefficient ratioImportant
(nr+1)(nr)=n−rr+1\frac{{n\choose r+1}}{{n\choose r}}=\frac{n-r}{r+1}
Variables
nn is a positive integer; 0≤r≤n−10\le r\le n-1.
Where used in JEE
Comparing terms, locating greatest term, solving coefficient inequalities.
Sum of all binomial coefficientsEssential
∑r=0n(nr)=2n\sum_{r=0}^{n}{n\choose r}=2^n
Variables
nn is a non-negative integer.
Where used in JEE
Coefficient sums, direct identities via putting \(x=a=1\).
Alternating sum of binomial coefficientsEssential
∑r=0n(−1)r(nr)=0(n≥1)\sum_{r=0}^{n}(-1)^r{n\choose r}=0\quad (n\ge 1)
Variables
nn is a positive integer.
Conditions
For n=0n=0, the sum equals 11.
Where used in JEE
Alternating coefficient sums via \((1-1)^n\), sign-based simplification.
Even and odd coefficient sumsImportant
∑r=0r evenn(nr)=∑r=0r oddn(nr)=2n−1(n≥1)\sum_{\substack{r=0\\ r\text{ even}}}^{n}{n\choose r}=\sum_{\substack{r=0\\ r\text{ odd}}}^{n}{n\choose r}=2^{n-1}\quad (n\ge 1)
Variables
nn is a positive integer.
Where used in JEE
Separating even and odd terms in binomial expansions.
Weighted sum of coefficientsImportant
∑r=0nr(nr)=n2n−1\sum_{r=0}^{n}r{n\choose r}=n2^{n-1}
Variables
nn is a positive integer.
Where used in JEE
Coefficient-weight sums, applications of differentiation of \((1+x)^n\).
Second weighted sum of coefficientsSupplementary
∑r=0nr(r−1)(nr)=n(n−1)2n−2\sum_{r=0}^{n}r(r-1){n\choose r}=n(n-1)2^{n-2}
Variables
nn is a positive integer.
Where used in JEE
Finding \(\sum r^2{n\choose r}\), advanced coefficient-sum questions.
Sum involving square of indexSupplementary
∑r=0nr2(nr)=n(n+1)2n−2\sum_{r=0}^{n}r^2{n\choose r}=n(n+1)2^{n-2}
Variables
nn is a positive integer.
Where used in JEE
Advanced weighted binomial coefficient sums.
Sum of squares of binomial coefficientsImportant
∑r=0n(nr)2=(2nn)\sum_{r=0}^{n}{n\choose r}^2={2n\choose n}
Variables
nn is a non-negative integer.
Where used in JEE
Standard identities, coefficient comparison in products.
Vandermonde-type identitySupplementary
∑r=0k(mr)(nk−r)=(m+nk)\sum_{r=0}^{k}{m\choose r}{n\choose k-r}={m+n\choose k}
Variables
m,nm,n are non-negative integers; kk is an integer.
Conditions
Terms with invalid lower indices are taken as zero; effectively 0≤k≤m+n0\le k\le m+n.
Where used in JEE
Coefficient extraction in products of binomials.
Expansion of \((1+x)^n\)Essential
(1+x)n=1+nx+n(n−1)2!x2+n(n−1)(n−2)3!x3+⋯+xn(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\cdots+x^n
Variables
nn is a positive integer.
Where used in JEE
Standard series-style expansion, approximation for small \(x\) when only first few terms are needed.

Expansion of \((1-x)^n\)

Essential
(1−x)n=1−nx+n(n−1)2!x2−n(n−1)(n−2)3!x3+⋯+(−1)nxn(1-x)^n=1-nx+\frac{n(n-1)}{2!}x^2-\frac{n(n-1)(n-2)}{3!}x^3+\cdots+(-1)^nx^n
Variables
nn is a positive integer.
Where used in JEE
Alternating expansions, coefficient sign questions.

General term and middle term and simple applications

General term in \((x+a)^n\)Essential
Tr+1=(nr)xn−rarT_{r+1}={n\choose r}x^{n-r}a^r
Variables
r=0,1,2,…,nr=0,1,2,\dots,n.
Conditions
nn is a positive integer.
Where used in JEE
Finding specific terms, coefficients, independent term, rational term.
General term in \((ax+b)^n\)Essential
Tr+1=(nr)(ax)n−rbr=(nr)an−rbrxn−rT_{r+1}={n\choose r}(ax)^{n-r}b^r={n\choose r}a^{n-r}b^r x^{n-r}
Variables
r=0,1,2,…,nr=0,1,2,\dots,n; a,ba,b constants.
Conditions
nn is a positive integer.
Where used in JEE
Coefficient of \(x^k\), term independent of \(x\), variable-power matching.
General term in \((a/x+b)^n\)Important
Tr+1=(nr)(ax)n−rbr=(nr)an−rbrxr−nT_{r+1}={n\choose r}\left(\frac{a}{x}\right)^{n-r}b^r={n\choose r}a^{n-r}b^r x^{r-n}
Variables
r=0,1,2,…,nr=0,1,2,\dots,n.
Conditions
x≠0x\ne 0.
Where used in JEE
Finding constant term, powers of \(x\), term with given exponent.
General term in \((x^p+a/x^q)^n\)Essential
Tr+1=(nr)(xp)n−r(axq)r=(nr)arxpn−(p+q)rT_{r+1}={n\choose r}(x^p)^{n-r}\left(\frac{a}{x^q}\right)^r={n\choose r}a^r x^{pn-(p+q)r}
Variables
p,qp,q are real numbers, usually integers; r=0,1,…,nr=0,1,\dots,n.
Conditions
x≠0x\ne 0 when negative powers occur.
Where used in JEE
Constant term and specified power of \(x\) in mixed-power binomials.
Specified power condition from general termImportant
If Tr+1 contains xk, then equate the exponent of x in Tr+1 to k.\text{If }T_{r+1}\text{ contains }x^{k},\text{ then equate the exponent of }x\text{ in }T_{r+1}\text{ to }k.
Variables
kk is the required exponent.
Where used in JEE
Finding a term independent of \(x\), coefficient of \(x^k\), or term containing \(x^k\).
Constant term conditionEssential
Constant term occurs when the exponent of x in Tr+1 is 0.\text{Constant term occurs when the exponent of }x\text{ in }T_{r+1}\text{ is }0.
Variables
rr is the term index parameter in the general term.
Conditions
Required rr must be an integer with 0≤r≤n0\le r\le n.
Where used in JEE
Independent term problems in expansions involving positive and negative powers.
Number of terms in binomial expansionImportant
Number of terms in (x+a)n is n+1\text{Number of terms in }(x+a)^n\text{ is }n+1
Variables
nn is a positive integer.
Conditions
Distinct terms considered in standard expansion order.
Where used in JEE
Middle term determination, counting arguments.
Middle term for even \(n\)Essential
If n is even, middle term is (n2+1)th term =Tn2+1\text{If }n\text{ is even, middle term is }\left(\frac{n}{2}+1\right)\text{th term }=T_{\frac{n}{2}+1}
Variables
nn is an even positive integer.
Where used in JEE
Finding the unique middle term in expansions with odd number of terms.
Middle terms for odd \(n\)Essential
If n is odd, middle terms are (n+12)th and (n+32)th terms\text{If }n\text{ is odd, middle terms are }\left(\frac{n+1}{2}\right)\text{th and }\left(\frac{n+3}{2}\right)\text{th terms}
Variables
nn is an odd positive integer.
Where used in JEE
Finding the two middle terms in expansions with even number of terms.
Middle term(s) in \((x+a)^n\)Important
For even n,  Tn2+1=(nn2)xn2an2\text{For even }n,\; T_{\frac{n}{2}+1}={n\choose \frac{n}{2}}x^{\frac{n}{2}}a^{\frac{n}{2}}
Variables
nn is an even positive integer.
Where used in JEE
Direct evaluation of middle term in standard binomial expansion.
Two middle terms in \((x+a)^n\)Important
For odd n,  Tn+12=(nn−12)xn+12an−12,Tn+32=(nn+12)xn−12an+12\text{For odd }n,\; T_{\frac{n+1}{2}}={n\choose \frac{n-1}{2}}x^{\frac{n+1}{2}}a^{\frac{n-1}{2}},\qquad T_{\frac{n+3}{2}}={n\choose \frac{n+1}{2}}x^{\frac{n-1}{2}}a^{\frac{n+1}{2}}
Variables
nn is an odd positive integer.
Where used in JEE
Direct evaluation of two middle terms.
Coefficient of \(x^{n-r}\) in \((x+a)^n\)Important
Coeff. of xn−r is (nr)ar\text{Coeff. of }x^{n-r}\text{ is }{n\choose r}a^r
Variables
r=0,1,…,nr=0,1,\dots,n.
Where used in JEE
Direct coefficient extraction in standard form.
Coefficient of \(x^k\) in \((ax+b)^n\)Essential
If k=n−r, then coeff. of xk is (nr)akbn−k=(nk)akbn−k\text{If }k=n-r,\text{ then coeff. of }x^k\text{ is }{n\choose r}a^k b^{n-k}={n\choose k}a^k b^{n-k}
Variables
0≤k≤n0\le k\le n.
Conditions
kk must be an integer.
Where used in JEE
Finding coefficients of a specified power of \(x\).
Independent term in \((x^p+a/x^q)^n\)Essential
Constant term exists when pn−(p+q)r=0  ⇒  r=pnp+q\text{Constant term exists when }pn-(p+q)r=0\;\Rightarrow\; r=\frac{pn}{p+q}
Variables
rr is the binomial term parameter.
Conditions
Constant term exists only if pnp+q\frac{pn}{p+q} is an integer in [0,n][0,n].
Where used in JEE
Determining independent term in mixed-power expansions.
Term independent of \(x\) in \((a/x+bx)^n\)Important
Constant term exists when 2r−n=0  ⇒  r=n2,Tn2+1=(nn2)an2bn2\text{Constant term exists when }2r-n=0\;\Rightarrow\; r=\frac{n}{2},\quad T_{\frac{n}{2}+1}={n\choose \frac{n}{2}}a^{\frac{n}{2}}b^{\frac{n}{2}}
Variables
nn is a positive integer.
Conditions
Exists only when nn is even.
Where used in JEE
Very common constant-term question.
Greatest binomial coefficient(s)Supplementary
max⁡{(n0),(n1),…,(nn)}={(nn/2),n even(n(n−1)/2)=(n(n+1)/2),n odd\max\{{n\choose 0},{n\choose 1},\dots,{n\choose n}\}=\begin{cases}{n\choose n/2},& n\text{ even}\\[4pt]{n\choose (n-1)/2}={n\choose (n+1)/2},& n\text{ odd}\end{cases}
Variables
nn is a non-negative integer.
Where used in JEE
Largest coefficient, middle coefficient questions.
Relation for increasing binomial coefficientsSupplementary
(nr+1)>(nr)  ⟺  r<n−12{n\choose r+1}>{n\choose r}\iff r<\frac{n-1}{2}
Variables
nn is a positive integer.
Conditions
Uses integer rr with 0≤r≤n−10\le r\le n-1.
Where used in JEE
Locating greatest coefficient or greatest term in symmetric expansions.

Popular questions in Binomial Theorem and Its Simple Applications

Frequently asked questions

What are the important Binomial Theorem and Its Simple Applications formulas for JEE?

This page lists 33+ JEE-relevant Binomial Theorem and Its Simple Applications formulas organised by subtopic. Start with essential formulas, then important identities before supplementary shortcuts.

Is this Binomial Theorem and Its Simple Applications formula sheet aligned with JEE Main?

Yes. Every formula is mapped to the JEE Main Mathematics syllabus for Binomial Theorem and Its Simple Applications, covering Binomial theorem for a positive integral index, General term and middle term and simple applications.

How should I revise the Binomial Theorem and Its Simple Applications formula sheet before JEE?

Revise essential formulas daily, important ones every 2–3 days, and supplementary results weekly. After each pass, solve 10–15 MCQs to test recall under exam conditions.

Where can I practise Binomial Theorem and Its Simple Applications MCQs after revising formulas?

Use the Online Practice or MCQs pages for the same unit on Goodmarks to convert formula recall into problem-solving speed.

Does this replace NCERT for Binomial Theorem and Its Simple Applications?

No — use this formula sheet for quick revision alongside NCERT and your coaching notes. Formulas here are a condensed reference, not a substitute for concept building.

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