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Binomial Expansion

(x + y)n = nC0xn + nC1xn–1y + nC2 xn–2 y2 + ... + nCnyn,
 
where n N
= 67185.png
  • This expansion has (n + 1) terms.
  • In general, the term is given by
     
    Tr+1 = nCr xnr yr
     
    where r = 0, 1, 2, 3, ... , n.
  • (p + 1)th term from end = (np + 1)th term from beginning = Tnp+1.

Properties of binomial coefficient

  • Sum of two consecutive binomial coefficient,
    nCr + nCr–1 = n+1Cr
  • r nCr = n n–1Cr–1
  • Ratio of two consecutive binomial coefficient,
    67179.png
  • If nCx = nCy, then either x = y or x + y = n
  • nC0 + nC1 + nC2 + ... + nCn = 2n
  • nC0nC1 + nC2 – ... nCn = 0
  • nC0 + nC2 + nC4 + ... = nC1 + nC3 + nC5 + ... = 2n–1
  • Putting x = i in (1 + x)n and comparing real and imaginary parts, we have
    nC0nC2 + nC4nC6 + ... = 67173.png
    and nC1nC3 + nC5 – ... = 2n/2 sin67167.png




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