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Cheat sheet · JEE Preparation Mathematics

Binomial Theorem cheat sheet

The whole chapter on one page: key formulas, facts and the points to remember, for quick revision before a test.

JEE · Mathematics

Cheat sheet

Class 11 · Chapter 7Binomial Theorem

Pascal's triangle gives the coefficients: each number is the sum of the two above.

1. Expansion

(a+b)n=∑r=0nnCr an−rbr(a + b)^n = \sum_{r = 0}^{n} {}^nC_r\,a^{n - r}b^r
Tr+1=nCr an−rbrT_{r + 1} = {}^nC_r\,a^{n - r}b^r

There are n + 1 terms; powers of a fall, powers of b rise.

2. Middle Term

  • n even: one middle term Tn2+1T_{\frac n2 + 1}.
  • n odd: two middle terms Tn+12T_{\frac{n + 1}{2}} and Tn+32T_{\frac{n + 3}{2}}.

3. Useful Results

 nC0+nC1+⋯+nCn=2n\,{}^nC_0 + {}^nC_1 + \cdots + {}^nC_n = 2^n

(1.1)10000>1000(1.1)^{10000} > 1000: (1+0.1)10000>1+10000(0.1)=1001(1 + 0.1)^{10000} > 1 + 10000(0.1) = 1001.

4. Key Points

  • Coefficients equidistant from ends are equal.
  • Term independent of x: set the power of x to 0.
  • (a − b)ⁿ: alternate signs.
  • Pascal: ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ.

Remember: General term T_{r+1} | middle term | sum = 2ⁿ

Binomial Theorem: questions and notes

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