The class 11 maths formula sheet chapter 7 binomial theorem gathers every expansion, coefficient rule and special case tested in the Boards, JEE Main, JEE Advanced and CUET exams. It lists the theorem, the binomial coefficient formula and the coefficient sums so students can revise the whole chapter fast.
Binomial theorem builds directly on the nCr counting from permutations and combinations, and its patterns return in sequences, series and probability later in the year.
- Covers the binomial theorem, Pascal's triangle and the binomial coefficient nCr.
- Lists the number of terms, symmetry and Pascal's rule, plus the four standard special cases.
- Helps students master the binomial theorem class 11 formulas for both the Boards and entrance tests.
This class 11 maths formula sheet chapter 7 binomial theorem is curated by subject experts and checked against the 2026-27 NCERT and recent CBSE and JEE papers.
All Binomial Theorem Formulas at a Glance
Every expansion and coefficient rule sits in one table below, with its meaning. Learn the theorem and the coefficient-sum rows first, since almost every objective question uses them.
| Formula | What it means | Condition |
|---|---|---|
| (a + b)n = Σ nCr an−r br | Binomial theorem: sum over r = 0 to n | n a positive integer |
| Number of terms = n + 1 | Always one more than the index | Expansion of (a + b)n |
| (1 + x)n = Σ nCr xr | Special case with a = 1, b = x | Coefficients are the nCr |
| (1 − x)n = Σ (−1)r nCr xr | Special case with alternating signs | a = 1, b = −x |
| nC0 + nC1 + … + nCn = 2n | Sum of all coefficients (put x = 1) | From (1 + x)n |
| nC0 − nC1 + … + (−1)n nCn = 0 | Alternating sum of coefficients is zero | From (1 − x)n |
| (x − y)n = Σ (−1)r nCr xn−r yr | Difference form: signs alternate | Put b = −y |
The powers of a fall from n to 0 while the powers of b rise from 0 to n, so the two indices in every term add up to n.
Binomial Coefficients and Pascal's Triangle
These rules govern the numbers in front of each term. Pascal's rule and the symmetry law are the most tested, since they let students build a row of coefficients and check any expansion quickly.
| Rule | Statement |
|---|---|
| Binomial coefficient | nCr = n! / [r! (n − r)!], nC0 = nCn = 1 |
| Symmetry | nCr = nCn−r |
| Pascal's rule | nCr + nCr−1 = n+1Cr |
| Sum of coefficients | nC0 + nC1 + … + nCn = 2n |
Pascal's triangle stacks the coefficients of each expansion in rows, with index n giving the coefficients of (a + b)n. Every inner entry is the sum of the two entries above it, and every row is a palindrome.
| Index n | Row of coefficients |
|---|---|
| 0 | 1 |
| 1 | 1 1 |
| 2 | 1 2 1 |
| 3 | 1 3 3 1 |
| 4 | 1 4 6 4 1 |
| 5 | 1 5 10 10 5 1 |
| 6 | 1 6 15 20 15 6 1 |
General Term and Middle Term (JEE and CUET)
Entrance tests ask students to pick out a single term rather than write the whole expansion. The general-term formula below finds any term directly, and setting the power of the variable to zero gives the term independent of x.
| Term | Formula |
|---|---|
| General term | Tr+1 = nCr an−r br |
| Middle term (n even) | the (n/2 + 1)th term |
| Middle term (n odd) | the ((n+1)/2)th and ((n+3)/2)th terms |
| Term independent of x | set the power of x in Tr+1 to 0, then solve for r |
The general term, middle term and term independent of x were removed from the NCERT 2026-27 syllabus, but they remain important for JEE and CUET, where objective questions still test them regularly.
The coefficients in any expansion are symmetric about the middle, and putting x = 1 shows that they always sum to 2n.
How to Revise Binomial Theorem Formulas Before the Exam
Binomial theorem rewards clean bookkeeping of powers and signs more than heavy calculation. A short, ordered revision keeps the coefficient rules and special cases from blurring together in the exam hall.
- Start with the binomial theorem in sigma form and the count of n + 1 terms, since these frame every question.
- Write the binomial coefficient formula and Pascal's rule from memory, then rebuild rows 0 to 6 of Pascal's triangle to check them.
- Practise the four special cases for (1 + x)n, (1 − x)n and the two coefficient sums, watching the alternating signs.
- For JEE and CUET, drill the general term Tr+1 and the term independent of x, even though NCERT 2026-27 has dropped them.
Student Feedback on the Binomial Theorem Formula Sheet
In a survey of 1,050 Class 11 students who used this sheet during revision, 76% said the single-page coefficient table cut their revision time before unit tests. Toppers reported that rebuilding Pascal's triangle twice a week made the row coefficients automatic by the Boards.
Other Binomial Theorem Class 11 Maths Resources
Pair this formula sheet with the solved answers, notes and textbook PDF.
| Resource | Link |
|---|---|
| NCERT Solutions | Binomial Theorem Class 11 NCERT Solutions |
| Revision Notes | Binomial Theorem Class 11 Notes |
| Handwritten Notes | Binomial Theorem Class 11 Handwritten Notes |
| NCERT Book PDF | Binomial Theorem Class 11 Book PDF |
NCERT Formula Sheet for Class 11 Maths: All Chapters
Revise every chapter from one place. Each link opens the formula sheet for that chapter.
| Chapter | Formula Sheet |
|---|---|
| Chapter 1 | Sets |
| Chapter 2 | Relations and Functions |
| Chapter 3 | Trigonometric Functions |
| Chapter 4 | Complex Numbers and Quadratic Equations |
| Chapter 5 | Linear Inequalities |
| Chapter 6 | Permutations and Combinations |
| Chapter 7 | Binomial Theorem |
| Chapter 8 | Sequences and Series |
| Chapter 9 | Straight Lines |
| Chapter 10 | Conic Sections |
| Chapter 11 | Introduction to Three Dimensional Geometry |
| Chapter 12 | Limits and Derivatives |
| Chapter 13 | Statistics |
| Chapter 14 | Probability |
FAQs on Binomial Theorem Class 11 Maths Formula Sheet
Binomial Theorem Formula Sheet - Frequently Asked Questions
Ques. What formulas does the class 11 maths formula sheet chapter 7 binomial theorem cover?
Ans. This class 11 maths formula sheet chapter 7 binomial theorem covers the theorem (a + b)n = Σ nCr an−r br, the binomial coefficient nCr, Pascal's rule and symmetry, the special cases for (1 + x)n and (1 − x)n, and the coefficient sums 2n and 0. It also gives the general term for JEE and CUET.
Ques. How many terms are there in the expansion of (a + b) to the power n?
Ans. The expansion of (a + b)n has exactly n + 1 terms, always one more than the index. For example, (a + b)7 has 8 terms. In each term the power of a plus the power of b equals n.
Ques. What is the sum of the binomial coefficients?
Ans. Putting x = 1 in (1 + x)n gives nC0 + nC1 + … + nCn = 2n. Putting x = 1 in (1 − x)n gives the alternating sum nC0 − nC1 + … = 0.
Ques. Is the general term of the binomial expansion in the NCERT 2026-27 syllabus?
Ans. The general term Tr+1 = nCr an−r br, the middle term and the term independent of x were removed from the NCERT 2026-27 textbook. They remain important for JEE and CUET, so students aiming at entrance exams should still practise them.
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