Binomial Theorem Class 11 Notes & Important Questions (Ch 7)

Revision notes

Binomial Theorem: revision notes

1. Pascal's triangle

The coefficients of $(a + b)^{n}$ for $n = 0, 1, 2, \dots$ form Pascal's triangle: each row starts and ends with $1$ , and every other entry is the sum of the two above it.

$1$ ; $1 1$ ; $1 2 1$ ; $1 3 3 1$ ; $1 4 6 4 1$ ; $1 5 10 10 5 1$ ; $1 6 15 20 15 6 1$ . Entry $r$ of row $n$ is $^{n} C_{r}$ (because $^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$ ).

2. The binomial theorem

For a positive integer $n$ :

$(a + b)^{n} =^{n} C_{0} a^{n} +^{n} C_{1} a^{n - 1} b +^{n} C_{2} a^{n - 2} b^{2} + \dots +^{n} C_{n} b^{n}$

There are $n + 1$ terms; powers of $a$ fall from $n$ to $0$ while powers of $b$ rise from $0$ to $n$ ; in each term the powers add up to $n$ .
For $(a - b)^{n}$ the signs alternate: $+, -, +, \dots$
$a = b = 1$ : $^{n} C_{0} +^{n} C_{1} + \dots +^{n} C_{n} = 2^{n}$ . Putting $x = 1$ in any expansion gives the sum of its coefficients.

3. Special cases

$(1 + x)^{n} = 1 + n x + \frac{n (n - 1)}{2} x^{2} + \dots + x^{n}$ . $(a + b)^{n} + (a - b)^{n}$ keeps only the terms with even powers of $b$ (doubled); $(a + b)^{n} - (a - b)^{n}$ only the odd ones.

4. Applications

Powers of numbers near a round number: $101^{3} = (100 + 1)^{3}$ , $(0.99)^{4} = (1 - 0.01)^{4}$ .
Divisibility: $(1 + k)^{n} = 1 + n k + k^{2} (\dots)$ , so $(1 + k)^{n} - 1 - n k$ is divisible by $k^{2}$ .
Comparing sizes: for $x > 0$ , $(1 + x)^{n} > 1 + n x$ ( $n \geq 2$ ).

Worked example 1

Expand $(1 + 2 x)^{3}$ .

$1 + 3 (2 x) + 3 (2 x)^{2} + (2 x)^{3} = 1 + 6 x + 12 x^{2} + 8 x^{3}$ .

Worked example 2

Evaluate $99^{3}$ using the binomial theorem.

$(100 - 1)^{3} = 1000000 - 30000 + 300 - 1 = 970299$ .

Worked example 3

Find $(\sqrt{2} + 1)^{4} + (\sqrt{2} - 1)^{4}$ .

$2 [(\sqrt{2})^{4} + 6 (\sqrt{2})^{2} + 1] = 2 (4 + 12 + 1) = 34$ .

Common errors

Forgetting to raise the coefficient of $x$ to the power: $(2 x)^{3} = 8 x^{3}$ , not $2 x^{3}$ .
Signs in $(a - b)^{n}$ : every odd power of $b$ carries a minus sign.
Using row $n - 1$ of Pascal's triangle for $(a + b)^{n}$ (the row starting $1, n$ is the right one).
Writing $n$ terms instead of $n + 1$ .

Exam tips

Write the coefficients $1, n,^{n} C_{2}, \dots$ first, then the powers of each part in brackets.
Check an expansion by putting $x = 1$ : both sides must agree.

Topics in this chapter: The binomial theorem · Pascal's triangle · Special cases · Applications.

Formulas for this chapter: Chapter 7 on the Class 11 formula sheet (free PDF and A3 poster).

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can use Pascal's triangle, count terms and find coefficients in small expansions.

Read first: 1. Pascal's triangle; 2. The binomial theorem 5 practice questions · checkpoint: 3 questions, 3 marks, pass 80%

Practise step 1

Step 2

Exam standard

You can expand binomials with coefficients and fractions and evaluate powers of numbers near round numbers.

Read first: 2. The binomial theorem; 4. Applications; Worked examples 1-3 6 practice questions · checkpoint: 3 questions, 8 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can prove divisibility results and combine (a + b)ⁿ ± (a − b)ⁿ in multi-part questions.

Read first: 3. Special cases; 4. Divisibility 3 practice questions · checkpoint: 2 questions, 9 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can compare sizes of powers and reason about sums of coefficients.

Read first: 4. Comparing sizes; 2. Sum of coefficients 3 practice questions · checkpoint: 3 questions, 9 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 2 of the 32 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceThe binomial theorem

The number of terms in the expansion of $(a + b)^{7}$ is

(a) $7$
(b) $8$
(c) $14$
(d) $6$

Q2·1 mark·Multiple choicePascal's triangle

The coefficient of $a^{2} b^{3}$ in $(a + b)^{5}$ is

(a) $5$
(b) $20$
(c) $15$
(d) $10$

Q3·1 mark·Multiple choiceThe binomial theorem

The sum of the coefficients in the expansion of $(1 + x)^{6}$ is

(a) $6$
(b) $36$
(c) $64$
(d) $32$

Worked solution·2 marks·Very short answerThe binomial theorem

Use the binomial theorem to write out ${(x - \frac{2}{x})}^{4}$ , $x \neq 0$ , term by term.

Full worked solution Free, no sign-in

$x^{4} + 4 x^{3} (- \frac{2}{x}) + 6 x^{2} \cdot \frac{4}{x^{2}}$ $+ 4 x (- \frac{8}{x^{3}}) + \frac{16}{x^{4}}$ M1
$= x^{4} - 8 x^{2} + 24 - \frac{32}{x^{2}} + \frac{16}{x^{4}}$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original combinatorics problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S62 · Problem S13 · Problem S61 · Problem S12. All 22 →

Knowledge organiser: Binomial Theorem on one page (key facts, formulas and mistakes to avoid; printable PDF)

Formulas in this topic: Binomial expansion. All formulas A–Z

Where marks are lost in Binomial Theorem

Coefficient inside the bracket not raised to the power: (3x)² written 3x². Fix: bracket every term before raising it.
Alternating signs missed in (a − b)ⁿ. Fix: the sign is (−1)^(power of b).
Wrong row of Pascal's triangle. Fix: (a + b)ⁿ uses the row 1, n, …, n, 1 with n + 1 entries.
Divisibility proofs that stop after expanding. Fix: take the common factor out and say what is left is an integer.
Size comparisons claimed without an inequality. Fix: keep the first three terms and show they already exceed the other number.

Test Binomial Theorem against the clock

Want it against the clock? Take a timed 30-mark chapter test on Binomial Theorem, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for JEE? Try the JEE-style questions on permutations, combinations and the binomial theorem.

Binomial Theorem: common questions

What is in CBSE Class 11 Maths Chapter 7, Binomial Theorem?

The topics are: The binomial theorem, Pascal's triangle, Special cases, Applications. This page has revision notes for each, and 32 board-style practice questions with step-marked solutions.

How many marks is Binomial Theorem worth?

It is in the Algebra unit, which carries 25 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Binomial Theorem?

Coefficient inside the bracket not raised to the power: (3x)² written 3x². Fix: bracket every term before raising it. Alternating signs missed in (a − b)ⁿ. Fix: the sign is (−1)^(power of b).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

Binomial Theorem Class 11: notes and important questions

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