Class 11th mathematics (binomial theorem)

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The binomial expression is an expression comprising of two terms connected by -ve or +ve sign. Equations like x + a, 2x – 3y, 1x−1x3, 7x−24x3 are examples of binomial expressions. The binomial expansion of (p+q)n will have a total of (n + 1) terms. The coefficients in the binomial expansion follow a pattern called as Pascal’s triangle. The sum of exponents of ‘p’ and ‘q’ is always equal to n.

Binomial Expression

[p + q]n = [ nC0 × pn ] + [ nC1 × (pn – 1) × q ] + [ nC2 × (pn – 2) × q2 ] + [ nC3 × (pn – 3 )× q3 ] + . . . . . . . . . . . . . + [ nCn – 1 × p × (qn – 1) ] + [ nCn × qn ]. Where, p and q are real numbers and n is a positive integer

⇒ Binomial Coefficient

The coefficients nC0, nC1, nC2 . . . . . . . . . nCn occurring in the binomial expression are called as Binomial coefficients. Given below are some conclusions that can be derived using the Binomial Theorem.
(i) [x + y]n = [ nC0 × (xn) ] + [ nC1 × (xn – 1) × y ] + [ nC2 × (xn – 2) × y2 ] + [ nC3 × (xn – 3) × y3 ] + . . . . . . . . . . . . . . . . . . + [ nCn × yn ]
(ii) [x – y]n = [ nC0 × (xn) ] – [ nC1 × (xn – 1) × y ] + [ nC2 × (xn – 2) × y2 ] – [ nC3 × (xn – 3) × y3 ] + . . . . . . . . . . . . . . . . . . +(-1)n [ nCn × yn ]
(iii) [1 – x]n = [ nC0 ] – [ nC1 . x ] + [ nC2 . x2 ] – [ nC3 . x3 ] + . . . . . . . . . . . . . . . . . . + (-1)n [ nCn . xn ]
(iv) (a + b)n = ∑nr=0 nCr (a)n – r × br
NOTE:
  1. nCr = n!r!(n−r)! where, n is a non-negative integer and [0 ≤ r ≤ n]
  2. nC0 = nCn = 1
  3. There are total (n + 1) terms in the expansion of (a + b)n

Important Formulas

  • The general term in the expansion of (a + b)n: Tr + 1 = nCr × (a)n – r × br
  • The middle term in the expansion of (a + b)n :
Case 1:
If n is even: The middle term = (n2+1)thterm
Case 2:
If n is odd: The middle term = (n+12)thtermand(n+12+1)thterm

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