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Binomial Theorem

A single-scroll CAT 2026 quant cheat sheet covering the Binomial Theorem end to end across 18 stacked boxes: expansion, general and middle terms, coefficient sums, the term independent of x, greatest coefficient, the negative/fractional-index series, and number-theory shortcuts for remainders and last-two-digit problems. Every concept box carries a worked numeric example, and the final three boxes give exam shortcuts, common traps, and a 30-second revision list.

6 mins referenceUpdated Jul 6, 2026

Binomial Theorem

CAT'26 QUANT CHEATSHEET
Every formula, term and remainder trick you need for CAT 2026 — on one page.

The Binomial Theorem turns up on CAT more often than aspirants expect — rarely as a standalone question, more often hidden inside a remainder problem, a coefficient hunt, or a quick middle-term trap under time pressure. Know the general term cold and most of these collapse into two lines of work. This cheat sheet gathers every formula you actually need: the expansion itself, general and middle terms, coefficient sums, the term independent of x, and the number-theory shortcuts (remainders and last-two-digit tricks) that examiners keep reusing. Every box carries a worked example with real numbers, because a formula you cannot apply under exam pressure is not worth memorising. It sits alongside the rest of Optima Learn's CAT preparation; once you start timing yourself, sanity-check your score with the CAT score predictor.

Binomial Theorem: every formula you need

1Binomial Expansion
Power of a two-term sum, expanded.
(a+b)n = Σk=0n C(n,k)·an−k·bk
Example: (x+2)3 = x3 + 3·x2·2 + 3·x·4 + 8 = x3 + 6x2 + 12x + 8.
CAT Insight: powers of a fall n→0, powers of b rise 0→n; the two exponents in every term add to n — a fast sanity check.
2General Term
Any single term of the expansion.
Tr+1 = C(n,r)·an−r·br
Example: 4th term of (2x+3)5, r=3: C(5,3)·(2x)2·33 = 10·4x2·27 = 1080x2.
Common Mistake: for the "kth term" use r = k−1, not r = k.
3Number of Terms
One more than the power.
Terms in (a+b)n = n + 1
Example: (a+b)10 has 10+1 = 11 terms.
CAT Insight: in (a+b)n+(a−b)n the odd terms cancel, leaving only ⌊n/2⌋+1 terms.
4Middle Term
Position depends on n even or odd.
n even → Tn/2 + 1  |  n odd → T(n+1)/2, T(n+3)/2
Example: (x+y)6, n=6 even → T4 = C(6,3)x3y3 = 20x3y3.
CAT Favourite: when a=b=1 the middle term carries the greatest binomial coefficient of the expansion.
5Sum of All Coefficients
Set all variables = 1.
Sum of coeffs of f(x) = f(1)
Example: (3x−2)4 at x=1 = (3−2)4 = 1.
CAT Hack: the constant term alone is f(0), so for (ax+b)n it is bn.
6Sum of Binomial Coefficients
Coefficients alone add to a power of 2.
C(n,0)+C(n,1)+…+C(n,n) = 2n
Example: n=5: 1+5+10+10+5+1 = 32 = 25.
CAT Hack: even-indexed and odd-indexed coefficients each sum to 2n−1 (set x=−1 in (1+x)n).
7Term Independent of x
Net power of x set to zero.
(net exponent of x in Tr+1) = 0 → r
Example: in (x2 + 1/x)6, Tr+1 = C(6,r)·x12−3r. Set 12−3r=0 → r=4, term = C(6,4) = 15.
CAT Hack: write the exponent of x as one linear expression in r first; then it is a single equation, not guesswork.
8Coefficient of a Power
Match the exponent.
Coeff of xm: solve exponent = m for r
Example: coeff of x4 in (2x−3)6: r=2 → C(6,2)·24·(−3)2 = 15·16·9 = 2160.
Common Mistake: do not drop the numeric parts 24 and (−3)2 — the coefficient is more than C(n,r).
9Greatest Coefficient
The middle one is largest.
n even → C(n, n/2); n odd → C(n, (n−1)/2)
Example: in (1+x)8, greatest coefficient = C(8,4) = 70.
CAT Insight: "greatest coefficient" ≠ "greatest term" — the term also depends on the value of x.
10Ratio of Consecutive Terms
A clean term-to-term ratio.
Tr+1 / Tr = ((n−r+1)/r)·(b/a)
Example: in (1+x)7, T3/T2 = ((7−2+1)/2)·x = 3x; at x=2 the ratio is 6.
CAT Hack: to find the greatest term, set Tr+1/Tr ≥ 1 and solve the inequality for r.
11Pascal's Triangle
Sum of the two above.
C(n,r) = C(n−1, r−1) + C(n−1, r)
Example: row n=4 is 1 4 6 4 1, since C(4,2) = C(3,1)+C(3,2) = 3+3 = 6.
CAT Hack: for small n, read coefficients straight off the triangle instead of computing factorials.
12Negative / Fractional Index
Infinite series when |x| < 1.
(1+x)n = 1 + nx + n(n−1)/2! x2 + n(n−1)(n−2)/3! x3 + …
Example: (1+x)−1 = 1 − x + x2 − x3 + … = 1/(1+x).
CAT Insight: no C(n,r) here — n is not a positive integer, so the series never terminates and needs |x| < 1.
13Remainders via Binomial
Split the base around the divisor.
(k·d + a)n ≡ an (mod d)
Example: 1723 ÷ 16: 17 = 16+1, so 1723 ≡ 123 = 1 (mod 16). Remainder = 1.
CAT Favourite: pick the split so the leftover is ±1 whenever possible — the whole expansion collapses to one term.
14Last Two Digits
Work mod 100.
(10k ± a)n: only the last two terms matter mod 100
Example: 2110 = (1+20)10 ≡ 1 + 10·20 ≡ 1 + 200 ≡ 1 (mod 100). Last two digits = 01.
CAT Hack: for a base ending in 1, tens digit of the answer = (tens of base × unit of exponent) mod 10.
15Terms in (a+b+c)n
More terms than n + 1.
Distinct terms = C(n+2, 2) = (n+1)(n+2)/2
Example: (a+b+c)4 → (5·6)/2 = 15 terms.
CAT Insight: a k-term power (x1+…+xk)n has C(n+k−1, k−1) terms.

CAT exam shortcuts, traps & revision

16CAT Exam Shortcuts
  • General term: Tr+1 = C(n,r)·an−r·br
  • Terms: n+1 for a binomial, (n+1)(n+2)/2 for a trinomial
  • Sum of binomial coeffs = 2n; even = odd = 2n−1
  • Constant term: set net power of x to 0, solve for r
  • Middle term carries the greatest coefficient
  • Remainder of an mod d: write base as (k·d ± 1)
17Most Common CAT Traps
  1. Off-by-one: the "kth term" needs r = k−1 in Tr+1.
  2. Confusing sum of coefficients (put x=1) with sum of binomial coefficients (2n).
  3. Forgetting the variable's coefficient: (2x+3)n ≠ (x+3)n.
  4. Using C(n,r) when n is negative or fractional — use the infinite series instead.
  5. Counting trinomial terms as n+1 rather than (n+1)(n+2)/2.
  6. Sign slips in (a−b)n: every odd-power term is negative.
1830-Second Revision
  • (a+b)n = Σ C(n,r)an−rbr
  • Terms = n+1; middle term = biggest coeff
  • ΣC(n,r) = 2n; halves = 2n−1
  • Constant term: net x-power = 0
  • Remainder: base = (multiple ± 1)n
  • Trinomial terms = (n+1)(n+2)/2

Binomial questions reward recognition over calculation — the sooner you spot "this is really a remainder in disguise," the more time you keep for the rest of the section. Drill this sheet until the general term is reflex, then test it on real sets. For more topic guides, browse the Optima Learn blog or explore every study guide, and work through the full CAT exam hub for section-wise strategy. When you want structured, mentor-led prep, the team at Optima Learn can map out your plan — book a free CAT 2026 call and line up your next eight weeks.

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