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Binomial Theorem revision notes

Every idea in the Binomial Theorem chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 27 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 concept, and "last" is the latest year one appeared. ★ marks ideas 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}{}^{n}C_r\,x^{n-r}a^{r}
nn a positive integer: n+1n+1 terms (not nn), so the nCr{}^nC_r average 2n/(n+1)2^n/(n+1). Powers in each term add to nn.
General term
6× asked, last 2025
Tr+1=nCr xn−rar,0≤r≤nT_{r+1}={}^{n}C_r\,x^{n-r}a^{r},\qquad 0\le r\le n
The kkth term has r=k−1r=k-1. Clear alog⁡ay=ya^{\log_a y}=y before expanding; for xlog⁡2xx^{\log_2 x} take log⁡2\log_2 of the equation.
  • Not NCk{}^NC_k for the kkth term. Counting starts at T1T_1 with r=0r=0, so TkT_k carries NCk−1{}^NC_{k-1}. The 30th and 12th terms of (1+x)N(1+x)^N carry NC29{}^NC_{29} and NC11{}^NC_{11}.
Term from the end
5× asked, last 2025
Tk′ of (a+b)n=Tk of (b+a)n,TkTk′=(ab)n−2k+2T'_k \text{ of } (a+b)^n = T_k \text{ of } (b+a)^n,\qquad \frac{T_k}{T'_k}=\left(\frac{a}{b}\right)^{n-2k+2}
Tk′T'_k = kkth from the end = (n−k+2)(n-k+2)th from the start. Terms whose rr add to nn share nCr{}^nC_r, so only powers differ.
Adding and subtracting 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+⋯ ](x+a)^n+(x-a)^n=2\,[T_1+T_3+T_5+\cdots],\qquad (x+a)^n-(x-a)^n=2\,[T_2+T_4+\cdots]
Only even (odd) powers of aa survive, so a=qa=\sqrt q clears. The + form has n/2+1n/2+1 terms (nn even), (n+1)/2(n+1)/2 (nn odd).

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Practise Binomial Theorem

23 more concepts in 6 topics

Coefficient of a given power; Middle term and consecutive coefficients; Sums of binomial coefficients; Remainders, divisibility and integer parts; and Multinomial expansion, plus ideas from factorials, series, number theory. 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 x^k21×
  • Term independent of x20×
  • Polynomial times a binomial6×
  • Product of (1 + x) and (1 − x) powers4×
  • Brackets that collapse5×
  • Sums with shifting powers8×
  • Rational and integral terms12×

Middle term and consecutive coefficients

  • Middle term8×
  • Coefficients of neighbouring rows7×
  • Ratio of consecutive coefficients14×

Sums of binomial coefficients

  • Sum of all coefficients8×
  • Even- and odd-index sums11×
  • Weighted sums: r and r² times nCr15×
  • Dividing by r + 1 (integration)9×
  • Alternating sums5×
  • Products of coefficients (Vandermonde)10×
  • Hockey stick (sum down a column)10×

Remainders, divisibility and integer parts

  • Remainder of a large power20×
  • Modulo a square and last digits4×
  • Integral part of (b + √a)ⁿ2×
  • Comparing large numbers1×

Multinomial expansion

  • Multinomial theorem6×
  • Coefficient in a trinomial power8×

Also asked here: factorials, series, number theory

  • Key points3

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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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