Class 11 Maths NCERT Solutions Chapter 7 Binomial Theorem | Boundless Maths
Chapter 7 Class 11 Maths NCERT Solutions · Unit II · Algebra

Class 11 Maths NCERT Solutions Chapter 7: Binomial Theorem

Free, step-by-step NCERT Solutions for the Binomial Theorem — expanding any power of a binomial without multiplying it out term by term, using Pascal's Triangle and the general-term formula — plus the Miscellaneous Exercise of divisibility proofs, expansion shortcuts, approximations, and trinomial expansions. Solved the way CBSE awards marks, with the key formulas and the mistakes that cost students marks every year, right on this page.

The Binomial Theorem answers a simple question in a powerful way: instead of multiplying (a + b) by itself n times to expand (a + b)ⁿ, is there a direct formula for every term? Pascal's Triangle already hints at the answer for small powers, and the Binomial Theorem generalises it into a single formula that works for any positive integer n. Once that formula is in hand, the chapter turns to its uses — finding one specific term without expanding the whole thing, proving divisibility results, approximating expressions like (1.02)⁶, and expanding expressions with three terms instead of two.

2Exercises (incl. Misc.)
20Total Questions
2026-27CBSE Syllabus
100%Solved

Class 11 Maths NCERT Solutions Chapter 7 — Overview

This Class 11 Maths NCERT Solutions Chapter 7 hub covers the Binomial Theorem, the chapter that gives a direct formula for expanding (a + b)ⁿ for any positive integer n, instead of multiplying it out step by step. The chapter opens with Pascal's Triangle, where each row's entries are exactly the coefficients that appear when (a + b)ⁿ is expanded, and shows how these coefficients — the binomial coefficients nCr — are built from the same combinatorial idea used in counting. The Binomial Theorem then states this pattern as a single formula: (a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ, summed over r from 0 to n.

Exercise 7.1 puts this formula to work — expanding binomials for a given power, computing specific powers like (101)⁴ or (99)⁵ by writing them as a binomial in disguise, and using the expansion to prove divisibility results. The Miscellaneous Exercise pushes further into the theorem's more demanding applications: finding a required term without expanding the whole binomial, comparing coefficients between two different expansions, using the first couple of terms of an expansion to approximate a number like (0.99)⁵, and expanding trinomials such as (2x + 3y + 4z)ⁿ by treating two of the three terms as a single binomial term first.

How the Chapter Builds

One Idea Leads to the Next

1

Pascal's Triangle

Where the coefficients of a binomial expansion come from, row by row. Exercise 7.1.

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2

The Binomial Theorem

(a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ — one formula for every power. Exercise 7.1.

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3

Computing Powers & Divisibility

Writing numbers like 101 as (100 + 1) to compute powers, and proving divisibility results. Exercise 7.1.

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4

Advanced Applications

The general term, approximations, and trinomial expansions. Misc. Exercise.

Quick Reference

Important Formulas — Chapter 7

Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below has the full printable version for all of the Binomial Theorem.

Binomial Theorem for Positive Integral Indices (§7.2)

Binomial Theorem

(a+b)^n=\displaystyle\sum_{r=0}^{n}\ {}^nC_r\,a^{n-r}b^r

Expands any power of a binomial directly, without repeated multiplication.

Binomial coefficient

{}^nC_r=\dfrac{n!}{r!\,(n-r)!}

The same coefficients that appear as entries in Pascal's Triangle, one row per value of n.

Expansion of (a − b)ⁿ

(a-b)^n=\displaystyle\sum_{r=0}^{n}(-1)^r\,{}^nC_r\,a^{n-r}b^r

Same formula as (a + b)ⁿ, but the sign alternates term by term — a very common place to slip up.

Computing Powers & Divisibility (§7.2)

Writing a number as a binomial

101^4=(100+1)^4

Splitting a number close to a round value into a sum makes the expansion mostly one large term plus a few small corrections.

Divisibility proofs

(1+x)^n=1+nx+\cdots+x^n

Expanding (1 + x)ⁿ and grouping every term except the first shows the whole expression is a multiple of x.

Sum of all binomial coefficients

{}^nC_0+{}^nC_1+\cdots+{}^nC_n=2^n

Follows immediately by substituting a = b = 1 into the Binomial Theorem.

General Term & Applications (Miscellaneous Exercise)

General term of an expansion

T_{r+1}={}^nC_r\,a^{n-r}b^r

Gives any single term of the expansion directly — note the term index is r + 1, not r.

Approximating a number

(1+x)^n\approx1+nx\ (x\text{ small})

Keeping only the first two terms of an expansion gives a fast, accurate estimate when x is small.

Expanding a trinomial

(a+b+c)^n=\big[(a+b)+c\big]^n

Group two of the three terms into one, expand with the Binomial Theorem, then expand each (a + b) power again.

Decision Guide

Which Binomial Theorem Technique Applies?

A quick way to decide, once you know what the question is actually asking for.

What the question is askingUse thisWhy
Expand (a + b)ⁿ or (a − b)ⁿ for a given nBinomial Theorem formulaWrite out each term nCr aⁿ⁻ʳbʳ, alternating signs if the binomial has a minus (§7.2).
Compute a power like (101)⁴ or (99)⁵Rewrite as a binomial firstSplit the number into a round value plus a small correction, e.g. 101 = 100 + 1, then expand (§7.2).
Prove an expression is divisible by some numberExpand and group termsWrite the expression as (1 + x)ⁿ or similar, expand, and show every term after the first shares the required factor (§7.2).
Find one specific term of an expansion, without expanding fullyGeneral term formulaUse T(r+1) = nCr aⁿ⁻ʳbʳ and solve for the value of r that gives the required term (Misc. Exercise).
Estimate a value like (0.99)⁵ or (1.02)⁶Keep only the first two or three termsWrite the number as (1 + x)ⁿ with x small, and drop the higher-power terms as negligible (Misc. Exercise).
Expand something with three terms, like (2x + 3y + 4z)ⁿGroup two terms, expand twiceTreat (2x + 3y) as one binomial term first, expand with the theorem, then expand each resulting power again (Misc. Exercise).
Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across both exercises.

  • Dropping the alternating signs in (a − b)ⁿ — every other term in the expansion of (a − b)ⁿ is negative; treating it exactly like (a + b)ⁿ and forgetting the (−1)ʳ factor is a very common slip.
  • Miscalculating nCr — forgetting the factorial in the denominator, or mixing up nCr with nPr, throws off every coefficient in the expansion that follows.
  • Confusing the term number with r — the general term T(r+1) is the (r+1)th term, not the rth; asking for the "5th term" means setting r = 4, not r = 5.
  • Forgetting the edge terms — nC0 and nCn both equal 1, and it's easy to accidentally drop or mis-simplify the first or last term of an expansion.
  • Choosing the wrong split for a divisibility proof — the binomial form used must actually match the number the question asks about divisibility by; picking an unrelated split makes the whole proof fall apart.
  • Losing track of which two terms are grouped in a trinomial expansion — in (a + b + c)ⁿ, once (a + b) is treated as a single term, every one of its own sub-expansions must still be carried through correctly in the second round.
Solve Chapter-Wise

Class 11 Maths NCERT Solutions Chapter 7 — Choose an Exercise

7.1

Exercise 7.1

Binomial Theorem for Positive Integral Indices — expansions, computing powers, and divisibility proofs · 14 questions

Solve Exercise 7.1 →
M

Miscellaneous Exercise

Divisibility proofs, expansion shortcuts, approximations, and trinomial expansions · 6 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

Every formula for the Binomial Theorem — plus every other Class 11 Maths chapter — in one printable PDF.

Get Formula Cards →
Common Questions

Frequently Asked Questions

Quick answers from Class 11 Maths NCERT Solutions Chapter 7, Binomial Theorem.

How many exercises are there in Chapter 7, Binomial Theorem?
There is one main exercise — 7.1 (Binomial Theorem for Positive Integral Indices, 14 questions) — plus a Miscellaneous Exercise of 6 questions covering divisibility proofs, expansion shortcuts, approximations, and trinomial expansions, totalling 20 questions.
What is the Binomial Theorem?
The Binomial Theorem gives a formula for expanding any power of a binomial (a + b)ⁿ, without multiplying it out term by term. It states that (a + b)ⁿ equals the sum, as r runs from 0 to n, of nCr multiplied by aⁿ⁻ʳ and bʳ, where nCr is the binomial coefficient — the same numbers that appear in Pascal's Triangle.
What is the general term in a binomial expansion?
The general term, usually written T(r+1), is the (r+1)th term of the expansion of (a + b)ⁿ and is given by T(r+1) = nCr aⁿ⁻ʳbʳ. It lets a specific term of an expansion be found directly, without writing out every earlier term.
Where can I find the official NCERT textbook for this chapter?
Binomial Theorem is Chapter 7 of the NCERT Class 11 Mathematics textbook, published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this page follow the exercises exactly as they appear there.
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