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NDA Binomial Theorem

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This page covers NDA Binomial Theorem with complete concept notes, 28 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Binomial Theorem— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

BINOMIAL THEOREM — NDA LEVEL ---

CORE CONCEPT The Binomial Theorem gives us a formula to expand expressions like (a + b)^n without multiplying them out step by step. Instead of doing (a + b)^10 by hand, we use a ready-made formula. This saves massive time in exams.

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2.Key Rules

Core rules you must know cold

The expansion of (a + b)^n has exactly (n + 1) terms.

The powers of 'a' decrease from n to 0. Powers of 'b' increase from 0 to n.

The sum of powers in every single term is always equal to n.

Coefficients follow Pascal's Triangle pattern.

All coefficients add up to 2^n (put a = 1, b = 1).

The expansion is valid for any positive integer n.

3.Formula Block

Memorise — at least one formula appears in every paper

(a + b)^n = C(n,0) * a^n * b^0 + C(n,1) * a^(n-1) * b^1 + ... + C(n,r) * a^(n-r) * b^r + ... + C(n,n) * b^n

T(r+1) = C(n, r) * a^(n-r) * b^r

Where C(n, r) = n! / (r! * (n-r)!)

- If n is EVEN: only one middle term = T(n/2 + 1)

- If n is ODD: two middle terms = T((n+1)/2) and T((n+3)/2)

General Formula:

General Term (this is the most tested formula):

Middle Term:

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4.Exam Patterns

What examiners ask — read before attempting PYQs

NDA regularly asks

- Find the general term T(r+1) - Find the middle term of an expansion - Find the coefficient of x^k in an expansion - Find which term is independent of x (no x in that term) - Find the sum of coefficients --- SHORTCUT / TRICK Trick 1 — SUM OF COEFFICIENTS: Put a = 1, b = 1 in the expansion. So sum of coefficients of (x + 1)^n = 2^n. Put x = 1

Trick 2 — TERM INDEPENDENT OF x

Set the power of x in the general term equal to zero and solve for r. That gives you the exact term number. No guessing needed. ---

5.Worked Example

Solve this step-by-step before moving on

1

Step 1

Write the general term. T(r+1) = C(6, r) * x^(6-r) * (1/x)^r

2

Step 2

Simplify the power of x. (1/x)^r = x^(-r) So power of x = (6 - r) + (-r) = 6 - 2r

3

Step 3

For the term independent of x, power of x = 0. 6 - 2r = 0 2r = 6 r = 3

4

Step 4

Find the term. T(3+1) = T(4) = C(6, 3) * x^0 = 20 Answer: The term independent of x is 20 (the 4th term). ---

6.Exam Traps

Common mistakes students make — avoid these

Students forget that r starts from 0, not 1. So T(r+1) is the term formula. If r = 3, the term is T(4), the 4th term — not the 3rd.

This error costs easy marks.

Key Points to Remember

The expansion of (a + b)^n has exactly (n + 1) terms total.
General Term formula: T(r+1) = C(n, r) * a^(n-r) * b^r — memorise this first.
Sum of all binomial coefficients in (1 + x)^n equals 2^n.
Middle term when n is even: T(n/2 + 1); when n is odd: two middle terms T((n+1)/2) and T((n+3)/2).
To find term independent of x: set power of x in general term = 0 and solve for r.
C(n, 0) = C(n, n) = 1 always; the first and last coefficients are always 1.
In (a + b)^n, sum of powers of a and b in every term is always equal to n.
C(n, r) = C(n, n-r) — coefficients are symmetric from both ends of the expansion.

Exam-Specific Tips

General term of (a + b)^n is T(r+1) = C(n, r) * a^(n-r) * b^r, where r = 0, 1, 2, ..., n.
Sum of all coefficients of (1 + x)^n = 2^n (obtained by substituting x = 1).
The number of terms in the expansion of (a + b)^n is always n + 1.
For even n, there is exactly ONE middle term: T(n/2 + 1).
For odd n, there are exactly TWO middle terms: T((n+1)/2) and T((n+3)/2).
C(n, r) = n! divided by r! multiplied by (n - r)! — this is the binomial coefficient formula.
In the expansion of (1 + x)^n, the coefficient of x^r is C(n, r).
The greatest coefficient in (1 + x)^n occurs at the middle term when n is even, and equals C(n, n/2).

Practice MCQs

Binomial Theorem — Practice Questions

28graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 28

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Practice 1easy

The middle term in the expansion of (x + 2)⁶ is:

Practice 2easy

In the binomial expansion of (2x + 3)⁵, what is the coefficient of x³?

Practice 3easy

The middle term in the binomial expansion of (x + 1/x)⁶ is:

Practice 4easy

If the coefficient of x⁴ in the expansion of (1 + x)ⁿ is 15, then n equals:

Practice 5easy

In the expansion of (2x - 3y)⁴, the term independent of x is:

Practice 6easy

The sum of the coefficients in the expansion of (3x - 2)⁵ is:

Practice 7easy

The coefficient of x⁵ in the expansion of (2 + x)⁸ is:

Practice 8easy

In the expansion of (1 + x)ⁿ, if the coefficient of x⁴ is 15, then n equals:

Practice 9easy

The middle term in the expansion of (x + 1/x)⁶ is:

Practice 10easy

The sum of the binomial coefficients in the expansion of (2x + 3y)⁵ is:

Practice 11easy

If the coefficient of x² in the expansion of (1 + x)ⁿ is 10, then the coefficient of x³ in the same expansion is:

Practice 12easy

The constant term in the expansion of (x² + 1/x)⁶ is:

Practice 13easy

If the coefficient of x⁴ in the expansion of (1 + x)ⁿ is 35, then n equals:

Practice 14easy

In the binomial expansion of (3x - 2y)⁴, the sum of all coefficients is:

Practice 15medium

In the expansion of (x + 1/x)¹⁰, find the constant term (independent of x).

Practice 16medium

The sum of the binomial coefficients in the expansion of (1 + x)ⁿ is 256. Find the value of n.

Practice 17medium

In the binomial expansion of (2x + 3)^5, find the coefficient of x^3.

Practice 18medium

The term independent of x in the expansion of (x + 1/x²)^9 is:

Practice 19medium

If the coefficient of x^7 in the expansion of (ax² + 1/bx)^10 is 120, find the value of ab.

Practice 20medium

In the binomial expansion of (1 + x)^n, if the coefficients of the 5th and 6th terms are equal, find n.

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60-Second Revision — Binomial Theorem

Formula: General term T(r+1) = C(n, r) * a^(n-r) * b^r — this is tested in almost every paper.
Remember: Number of terms in expansion = n + 1, not n.
Trick: To find term independent of x, set power of x in T(r+1) equal to zero and solve for r.
Remember: r starts from 0. So if r = 3, the answer is the 4th term (T4), not the 3rd.
Formula: Sum of coefficients = substitute x = 1 in the expression. Result = 2^n for (1+x)^n.
Trap: Middle term rule changes based on whether n is even or odd — check n first before solving.
Remember: C(n, r) = C(n, n-r). Use this to simplify large combinations quickly in MCQs.

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