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Binomial Theorem formula sheet

Every formula in the Binomial Theorem chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 28 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that formula, and "last" is the latest year one appeared. ★ marks formulas asked 8 or more times.

Expansion and general term

Binomial theorem
1× asked, last 2025
(x+a)n=∑r=0nnCr xn−rar(x+a)^n=\sum_{r=0}^n{}^nC_r\,x^{n-r}a^r
n+1n+1 terms; powers in each term add to nn.
General term
6× asked, last 2025
Tr+1=nCr xn−rar,0≤r≤nT_{r+1}={}^nC_r\,x^{n-r}a^r,\qquad 0\le r\le n
The kkth term has r=k−1r=k-1 (carries nCk−1{}^nC_{k-1}). Clear alog⁡ay=ya^{\log_a y}=y before expanding.
Term from the end
5× asked, last 2025
Tk′ of (a+b)n=Tk of (b+a)n,TkTk′=(ab)n−2k+2T^\prime_k\text{ of }(a+b)^n=T_k\text{ of }(b+a)^n,\qquad \frac{T_k}{T^\prime_k}=\left(\frac ab\right)^{n-2k+2}
kkth from the end = (n−k+2)(n-k+2)th from the start.
Adding the conjugate
3× asked, last 2025
(x+a)n+(x−a)n=2[T1+T3+T5+⋯ ](x+a)n−(x−a)n=2[T2+T4+⋯ ]\begin{aligned}&(x+a)^n+(x-a)^n=2[T_1+T_3+T_5+\cdots]\\&(x+a)^n-(x-a)^n=2[T_2+T_4+\cdots]\end{aligned}
Only even (odd) powers of aa survive; a=qa=\sqrt q clears. + form: n/2+1n/2+1 terms (nn even).

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Practise Binomial Theorem

24 more formulas in 5 topics

Coefficient of a given power; Middle term and consecutive coefficients; Sums of binomial coefficients; Remainders, divisibility and integer parts; and Multinomial expansion. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Coefficient of a given power

  • Coefficient of xᵏ21×
  • Term independent of x20×
  • Polynomial times a binomial6×
  • (1 + x)ᵖ(1 − x)^q4×
  • Collapsing brackets5×
  • Shifting powers8×
  • Rational terms12×

Middle term and consecutive coefficients

  • Middle term8×
  • Neighbouring rows7×
  • Consecutive coefficients14×
  • Greatest term

Sums of binomial coefficients

  • Sum of coefficients8×
  • Even and odd sums11×
  • Weighted sums15×
  • Dividing by r + 19×
  • Alternating sums5×
  • Vandermonde10×
  • Hockey stick10×

Remainders, divisibility and integer parts

  • Remainder of a large power20×
  • Modulo a square4×
  • Integral part of (b + √a)ⁿ2×
  • Comparing large numbers1×

Multinomial expansion

  • Multinomial theorem6×
  • Trinomial coefficient8×

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About 200 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

Practise Binomial Theorem

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