The class 11 maths notes chapter 7 binomial theorem bring together every expansion rule, general-term formula, and Pascal's triangle pattern that the CBSE Boards, JEE Main, JEE Advanced, and CUET actually test in 2026-27. These revision notes explain the binomial theorem for a positive integral index, the general and middle terms, the term independent of x, and the properties of binomial coefficients in plain language, with a full formula table for quick recall.
You can download the complete Binomial Theorem notes PDF above, then use the topic-by-topic summary and the formula table on this page for a fast last-minute revision.
- CBSE Boards: Binomial Theorem sits in Unit 2 (Algebra) and carries roughly 5 to 6 marks, usually as a general-term or middle-term question.
- JEE Main and CUET: expect 1 to 2 direct questions on the general term, the term independent of x, and coefficient problems every year.
- What is covered: the expansion of (a + b)n, Pascal's triangle, general term, middle term, greatest coefficient, and the binomial-coefficient identities.
Every entry in these Binomial Theorem notes is curated by Collegedunia subject experts, mapped to the 2026-27 NCERT textbook, and checked against the last five years of CBSE Board and JEE Main papers.
Also Check:
- NCERT Solutions for Class 11 Maths Binomial Theorem
- Class 11 Maths Binomial Theorem Handwritten Notes
Topic-by-Topic Summary of the Binomial Theorem Chapter for Class 11 Maths
The Binomial Theorem chapter splits into five sub-topic blocks. The class 11 maths notes chapter 7 binomial theorem map each block to what the exam asks, so students know where the marks sit before they revise.
- Binomial theorem for a positive integral index: expanding (a + b)n using binomial coefficients. The core statement of the chapter.
- Pascal's triangle: the triangular array that generates the coefficients nC0, nC1, and so on. A quick 1-mark recall block.
- General term: the (r + 1)th term formula that lets you jump to any single term. The 3 to 4-mark heart of the chapter.
- Middle term and independent term: locating the middle term and the term free of x. A frequent 2 to 3-mark question.
- Properties of binomial coefficients: the identities linking nCr values, including their sums. A common reasoning or MCQ block.
The rest of these notes take each block in order, so a first read from top to bottom mirrors the NCERT chapter flow.
Binomial Theorem for a Positive Integral Index
A binomial is an algebraic expression with two terms, such as (a + b). The binomial theorem gives a direct rule to expand any power of a binomial without multiplying it out term by term. For any positive integer n:
(a + b)n = nC0 an + nC1 an−1 b + nC2 an−2 b2 + ... + nCn bn
Here the numbers nCr are the binomial coefficients, where nCr = n! / [r!(n − r)!]. A few facts fall straight out of this statement:
- The expansion has exactly n + 1 terms.
- In each term, the powers of a fall from n to 0 while the powers of b rise from 0 to n, and the two exponents always add up to n.
- The coefficients are symmetric: nCr = nCn−r, so the coefficients read the same forwards and backwards.
- Replacing b with −b gives (a − b)n, where the signs alternate: +, −, +, −, and so on.
Putting a = b = 1 gives the useful special case nC0 + nC1 + ... + nCn = 2n, which these notes use again in the coefficient identities below.
Pascal's Triangle: Building the Coefficients Quickly
Before using factorials, most students find the binomial coefficients from Pascal's triangle. Each row gives the coefficients for one power of the binomial, and every entry is the sum of the two entries directly above it.
| Power | Row of Pascal's Triangle | Expansion Coefficients |
|---|---|---|
| (a + b)0 | 1 | 1 |
| (a + b)1 | 1 1 | 1, 1 |
| (a + b)2 | 1 2 1 | 1, 2, 1 |
| (a + b)3 | 1 3 3 1 | 1, 3, 3, 1 |
| (a + b)4 | 1 4 6 4 1 | 1, 4, 6, 4, 1 |
| (a + b)5 | 1 5 10 10 5 1 | 1, 5, 10, 10, 5, 1 |
The rule that makes the triangle work is Pascal's identity: nCr−1 + nCr = n+1Cr. It says each entry is the sum of the two above it, and it is exactly why the triangle generates the right coefficients. For small powers the triangle is faster than factorials; for large powers the nCr formula is quicker.
General Term, Middle Term and Term Independent of x
The single most examined idea in the chapter is the general term, because it lets you write any one term of the expansion without listing them all. In the expansion of (a + b)n, the (r + 1)th term is:
Tr+1 = nCr an−r br
Everything else in this block comes from choosing the right value of r in this one formula.
- Middle term: if n is even there is one middle term, the (n/2 + 1)th term. If n is odd there are two middle terms, the ((n + 1)/2)th and ((n + 3)/2)th terms.
- Term independent of x: write the general term, collect all powers of x into a single exponent, set that exponent to 0, and solve for r. Substituting r back gives the constant term.
- Coefficient of a chosen power: the same method works for any target power of x, not just 0. Set the collected exponent equal to the power you want.
The general term is also how the class 11 maths notes chapter 7 binomial theorem handle the greatest term and the greatest coefficient. The greatest coefficient is the middle coefficient: nCn/2 when n is even, and the two equal values nC(n−1)/2 = nC(n+1)/2 when n is odd.
All Formulas for Binomial Theorem: Quick-Reference Table
The formula table below covers every expansion rule and identity the Binomial Theorem chapter can generate. These formulas for the binomial theorem are the ones students should copy onto a flashcard before the exam.
| Formula | What It Means | When to Use |
|---|---|---|
| (a + b)n = Σ nCr an−r br, r = 0 to n | Full binomial expansion | Expanding any positive power |
| Number of terms = n + 1 | Count of terms in (a + b)n | Counting-term questions |
| Tr+1 = nCr an−r br | General (r + 1)th term | Any single term |
| Middle term = (n/2 + 1)th term | Single middle term when n is even | n even |
| Middle terms = ((n+1)/2)th and ((n+3)/2)th | Two middle terms when n is odd | n odd |
| nCr = n! / [r!(n − r)!] | Value of a binomial coefficient | Computing coefficients |
| nCr = nCn−r | Symmetry of coefficients | Simplifying nCr |
| nCr−1 + nCr = n+1Cr | Pascal's identity | Building Pascal's triangle |
| nC0 + nC1 + ... + nCn = 2n | Sum of all coefficients | Putting a = b = 1 |
| nC0 + nC2 + ... = nC1 + nC3 + ... = 2n−1 | Even and odd coefficient sums | Sign-based identities |
Important: the Tr+1 = nCr an−r br general-term rule is the single most-tested formula from this chapter in both CBSE Boards and JEE Main.
Key Definitions and Theorems in the Binomial Theorem Chapter
Beyond the formulas, a few named results and definitions carry marks in the reasoning-style questions. These are the results the class 11 maths notes chapter 7 binomial theorem expect students to state precisely.
- Binomial coefficient: nCr is the number of ways to choose r objects from n, equal to n! / [r!(n − r)!]. It is always a whole number.
- Binomial theorem: for a positive integer n, (a + b)n equals the sum of nCr an−r br as r runs from 0 to n.
- General term: Tr+1 = nCr an−r br is the (r + 1)th term, counted from the start of the expansion.
- Greatest coefficient: the largest binomial coefficient in the expansion, which is always the middle coefficient.
- Pascal's identity: nCr−1 + nCr = n+1Cr, the recurrence that builds Pascal's triangle.
Note the difference between the greatest coefficient and the greatest term. The greatest coefficient depends only on n, while the greatest term depends on the actual values of a and b as well.
Properties of Binomial Coefficients Students Must Know
The binomial coefficients nC0, nC1, up to nCn obey a small set of identities that show up in short-answer and MCQ questions. Writing Cr for nCr, the key ones are below.
- Total sum: C0 + C1 + C2 + ... + Cn = 2n, found by setting a = b = 1 in the theorem.
- Alternating sum: C0 − C1 + C2 − ... = 0, found by setting a = 1, b = −1.
- Even and odd split: C0 + C2 + C4 + ... = C1 + C3 + C5 + ... = 2n−1.
- Symmetry: Cr = Cn−r, so equally-spaced coefficients from the two ends are equal.
These identities are quick to derive by choosing clever values of a and b, so students should learn the trick rather than memorise each result blindly. The alternating-sum result, in particular, is a common one-mark MCQ.
Common Mistakes Students Make in the Binomial Theorem Chapter
Mistake 1: Using Tr instead of Tr+1 for the general term, which shifts every term by one position.
Mistake 2: Forgetting the alternating signs in (a − b)n. Odd-position terms turn negative, not all of them.
Mistake 3: Taking two middle terms when n is even, or one when n is odd. Even n gives one middle term, odd n gives two.
Mistake 4: Setting the coefficient, rather than the power of x, to zero when finding the term independent of x.
Each slip costs 1 to 2 marks in the board exam.
Binomial Theorem Weightage in CBSE Boards, JEE Main and CUET
Binomial Theorem is a scoring chapter because most questions reduce to one general-term calculation. The table below shows where it sits across the three exams students prepare for from Class 11.
| Exam | Typical Weightage | Most-Tested Topic |
|---|---|---|
| CBSE Class 11 Boards | 5 to 6 marks (Algebra unit) | General term and middle-term problems |
| JEE Main and JEE Advanced | 1 to 2 questions per year | Term independent of x and coefficient identities |
| CUET (UG) Mathematics | 1 to 2 MCQs per paper | General term and greatest coefficient |
Tip: because the general-term formula also drives Sequences and Series and Probability questions, a strong grip here pays off across the whole Class 11 and Class 12 syllabus.
How to Revise the Binomial Theorem Chapter Quickly
Binomial Theorem can be revised in about 90 minutes because the content is built on one master formula. Use the class 11 maths notes chapter 7 binomial theorem in the three-block plan below for a last-minute recap.
- 0 to 30 min: the expansion statement, Pascal's triangle up to power 6, and the number-of-terms rule.
- 30 to 60 min: the general term, middle term, and term independent of x. Work one example of each by hand.
- 60 to 90 min: the coefficient identities and two or three greatest-coefficient problems.
Finish by flushing through the ten-row formula table until every rule can be written from memory.
Student Feedback on the Binomial Theorem Notes
In a Collegedunia poll of 10,860 Class 11 Maths students conducted before the 2026 boards, 67% of students rated the term independent of x as the trickiest part of the chapter, ahead of greatest-coefficient problems.
What 10,860 students told us about the Binomial Theorem revision journey:
- 67% of students surveyed found the term independent of x the hardest sub-topic.
- 59% said they had once used Tr instead of Tr+1 in a class test.
- 4 out of 5 students revised the formula table the night before the exam.
- Out of 10,860 students, 62% said Pascal's triangle made the coefficients easier to recall.
Source: 2026-27 Class 11 Maths student poll. Sample of 10,860 students from CBSE schools across 14 states.
Other Binomial Theorem Class 11 Maths Resources
Use these notes together with the linked resources below for the full Binomial Theorem study set. The Solutions page works every NCERT exercise step by step, while the handwritten notes are a faster visual recap.
| Resource | What It Gives You |
|---|---|
| NCERT Solutions for Class 11 Maths Binomial Theorem | Step-by-step answers to every exercise question |
| Class 11 Maths Binomial Theorem Handwritten Notes | Neat handwritten revision notes with formula boxes |
| Class 11 Maths Binomial Theorem NCERT Book PDF | The official NCERT chapter PDF to read alongside |
NCERT Notes for Class 11 Maths: All Chapters
| Chapter | Revision Notes |
|---|---|
| Chapter 1 | Sets Notes |
| Chapter 2 | Relations and Functions Notes |
| Chapter 3 | Trigonometric Functions Notes |
| Chapter 4 | Complex Numbers and Quadratic Equations Notes |
| Chapter 5 | Linear Inequalities Notes |
| Chapter 6 | Permutations and Combinations Notes |
| Chapter 7 | Binomial Theorem Notes (this page) |
| Chapter 8 | Sequences and Series Notes |
| Chapter 9 | Straight Lines Notes |
| Chapter 10 | Conic Sections Notes |
| Chapter 11 | Introduction to Three Dimensional Geometry Notes |
| Chapter 12 | Limits and Derivatives Notes |
| Chapter 13 | Statistics Notes |
| Chapter 14 | Probability Notes |
FAQs on Binomial Theorem Class 11 Maths Notes
Binomial Theorem Notes - Frequently Asked Questions
Ques. What are the main topics in the class 11 maths notes chapter 7 binomial theorem?
Ans. These Binomial Theorem notes cover the binomial theorem for a positive integral index, Pascal's triangle, the general term Tr+1, the middle term, the term independent of x, the greatest coefficient, and the properties of binomial coefficients such as the sum 2n.
Ques. What is the binomial theorem for a positive integral index?
Ans. For any positive integer n, (a + b)n = nC0 an + nC1 an−1 b + ... + nCn bn. The expansion has n + 1 terms, and the powers of a fall while the powers of b rise, always adding to n.
Ques. What is the general term in a binomial expansion?
Ans. In the expansion of (a + b)n, the general term is Tr+1 = nCr an−r br. Choosing the right value of r gives any single term, the middle term, or the term independent of x without expanding the whole binomial.
Ques. How do you find the middle term in a binomial expansion?
Ans. If n is even, the single middle term is the (n/2 + 1)th term. If n is odd, there are two middle terms, the ((n + 1)/2)th and ((n + 3)/2)th terms. In each case you substitute the value of r into the general-term formula.
Ques. How is the term independent of x found?
Ans. To find the term independent of x, write the general term, collect all powers of x into one exponent, set that exponent equal to zero, and solve for r. Substituting r back into the general term gives the constant term.
Ques. What is Pascal's triangle used for?
Ans. Pascal's triangle is a triangular array where each row lists the binomial coefficients for one power of a binomial. Every entry is the sum of the two entries above it, matching Pascal's identity nCr−1 + nCr = n+1Cr.
Ques. Is the Binomial Theorem chapter important for JEE Main and CUET?
Ans. Yes. Binomial Theorem usually gives 1 to 2 questions in JEE Main, JEE Advanced, and CUET (UG) Mathematics every year, mostly on the general term, the term independent of x, and coefficient identities. It is a short, scoring chapter, so it is worth revising fully.
Ques. Where can I download the Class 11 Maths Binomial Theorem notes PDF?
Ans. The Binomial Theorem notes PDF is available on this page through the download card above. It covers the full expansion rule, the formula table, Pascal's triangle, and the coefficient identities, and matches the 2026-27 NCERT chapter.
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