CAT Quant · Study & Practice

Binomial Theorem

AreaModern Maths DifficultyModerate CAT weightage0–2 questions (more frequent in XAT/SNAP; often fused with Number System or P&C)

The Binomial Theorem gives you the full expansion of (a+b) raised to any positive integer power without multiplying the bracket out term by term. Instead of grinding through (a+b)⁵ by hand, you read off each term from a single template: T(r+1) = nCr · a^(n−r) · b^r. That one line is the engine behind almost every binomial question CAT, XAT and SNAP have asked — finding a specific term, the middle term, the term free of x, the greatest coefficient, or the sum of all coefficients. The chapter matters beyond its own questions too: the nCr machinery overlaps with Permutations & Combinations, the "term independent of x" idea reappears in algebra, and the sum-of-coefficients trick (just substitute the variables = 1) is a recurring shortcut. CAT tends to test understanding over brute force — it rewards students who know exactly which term they need and jump straight to it rather than expanding everything. This chapter builds that aim-and-fire skill: the expansion pattern first, then the general term as a universal tool, then the high-value special cases, each with worked examples, the fastest method, and the traps that quietly cost marks.

Topics

1 Expansion 4 worked · 5 practice · 8-Q test Start → 2 General Term 4 worked · 5 practice · 8-Q test Start → 3 Middle Term 4 worked · 5 practice · 8-Q test Start → 4 Coefficients 4 worked · 5 practice · 8-Q test Start →

⚡ CAT shortcuts & speed methods

The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.

  • Never expand to find one term — set up T(r+1) = nCr·a^(n−r)·b^r and solve for r directly.
  • For "term containing x^k", collect every power of x into one exponent, set it = k, solve for r, then read the coefficient.
  • Sum of ALL coefficients: substitute every variable = 1 ⇒ (sum of the numerical bases)ⁿ. Sum of pure binomial coefficients = 2ⁿ.
  • Alternating sum nC0 − nC1 + nC2 − … = (1−1)ⁿ = 0 for n ≥ 1 — a one-line answer.
  • Middle term: n even ⇒ one term at r = n/2; n odd ⇒ two terms at r = (n−1)/2 and (n+1)/2. Fix parity first.
  • Greatest binomial coefficient sits dead centre: nC(n/2) for even n; the equal pair nC((n−1)/2) for odd n.

⚠️ Common mistakes & traps

CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.

  • Mixing up term number and r — the (r+1)th term uses r, so the 4th term means r = 3, not r = 4.
  • Treating +x% style signs wrong: in (a − b)ⁿ the (−1)^r factor flips odd-r terms — dropping the sign loses the answer.
  • Assuming the middle term is automatically the term independent of x — compute the power of x; it often is not.
  • Forgetting (a+b)ⁿ has (n+1) terms, so miscounting the middle position when n is odd vs even.
  • For "sum of coefficients", students substitute x = 0 instead of x = 1 — that gives only the constant term, not the sum.

📈 CAT exam insight & PYQ analysis

Binomial Theorem is a low-frequency but reliable CAT topic and a more regular XAT/SNAP guest. When it appears, the favourites are: finding a specific term or its coefficient (often the term independent of x in something like (x²+1/x)ⁿ), the middle term, and the sum/greatest of coefficients via clever substitution. Pure expansion is almost never asked; the questions test whether you can jump to the right r without writing the whole series. Difficulty is Easy–Moderate when the binomial is clean and Moderate–Hard when fused with Number System (remainders, last digits using (10±k)ⁿ) or P&C. Prioritise the general-term method and the substitution trick — they cover the bulk of what has actually been set.

🎴 Flashcards — instant recall

Tap a card to reveal the answer. Drill these until they are automatic.

General term of (a+b)ⁿ?Tap to reveal
T(r+1) = nCr · a^(n−r) · b^r
Number of terms in (a+b)ⁿ?Tap to reveal
n + 1
Which r gives the 6th term?Tap to reveal
r = 5 (term number = r + 1)
Sum of all binomial coefficients nC0+…+nCn?Tap to reveal
2ⁿ
Alternating sum nC0 − nC1 + nC2 − …?Tap to reveal
0 (for n ≥ 1)
Sum of ALL coefficients in an expansion — how?Tap to reveal
Put every variable = 1
Middle term when n is even?Tap to reveal
One term, T(n/2 + 1)
Middle terms when n is odd?Tap to reveal
Two: T((n+1)/2) and T((n+3)/2)
Greatest binomial coefficient (n even)?Tap to reveal
nC(n/2), the central one
Sign of the r-th term in (a−b)ⁿ?Tap to reveal
(−1)^r
Symmetry property of nCr?Tap to reveal
nCr = nC(n−r)
To get the term free of x, set what equal to zero?Tap to reveal
The combined exponent of x in T(r+1)

📌 Quick revision

Binomial Theorem expands (a+b)ⁿ as Σ nCr·a^(n−r)·b^r with (n+1) terms whose exponents sum to n. The general term T(r+1) = nCr·a^(n−r)·b^r is the workhorse: solve for r to find any specific term, the term containing x^k, or the term free of x (set the combined power of x to 0). Middle term: n even ⇒ one term at r = n/2; n odd ⇒ two terms. The sum of all coefficients comes from putting variables = 1; pure binomial coefficients sum to 2ⁿ and alternate-sum to 0. The greatest coefficient is the central one. Watch the term-number-vs-r off-by-one and the (−1)^r sign in (a−b)ⁿ.

Chapter test

📝 Binomial Theorem — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
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🏆 Vidaara CAT success checklist

You have truly mastered Binomial Theorem when you can tick every box below.

  • Recall every formula in this chapter without looking them up
  • Solve each topic’s practice set with at least 80% accuracy
  • Use the chapter shortcuts to cut your solving time in half
  • Spot and avoid every common trap listed above
  • Score 80%+ on the timed chapter test

📋 Chapter mastery scorecard

Track where you stand. Aim for the target before moving to the next chapter.

Skill checkpointTarget
Concept theory & formulas understood100%
Topic practice sets attempted (4 topics)4/4
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall12 cards