Binomial Theorem Class 11 Notes - Chapter 8

The binomial expression is an expression comprising of two terms connected by -ve or +ve sign. Equations like x + a, 2x – 3y, \(\frac{1}{x}-\frac{1}{x^{3}}\), \(7x-\frac{2}{4x^{3}}\) 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

\(\Rightarrow\) 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 = \(\sum_{r\;=\;0}^{n}\) nCr (a)n – r × br

NOTE:

  1. nCr = \(\frac{n!}{r!(n-r)!}\) 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 = \(\left ( \frac{n}{2}+1 \right )^{th}term\)

Case 2:

If n is odd: The middle term = \(\left ( \frac{n+1}{2}\right )^{th}term\;\;and\;\;\left ( \frac{n+1}{2} +1 \right )^{th}term\)

Binomial Theorem Class 11 Important Questions

  1. Determine the coefficient of \((x)^{n}\) in the expansion of \((x^{3}+3x^{2}+4x-17)^{4}\)
  2. If the coefficient of 2nd, 3rd and 4th terms in the expansion of \((1+x)^{2n}\) are in Arithmetic Progression. Show that \(2n^{2}-9n+7=0\).
  3. Find the greatest term in the expansion of \((2+3x)^{9}\), where x = 3/2.
  4. Determine the 4th term from the end in the expansion of \(\left [ \frac{x^{2}}{5}-\frac{x}{3} \right ]^{8}\)
  5. Expand the following \(\left [2x^{2}-5x+8\right ]^{4}\)

Also Read


Practise This Question

Mean of 100 items is 49. It was discovered that three items which should have been 60,70,80 were wrongly read as 40, 20, 50 respectively.  The correct mean is