Binomial Theorem: Maths Class 11 Chapter-8

Key Features of NCERT Material for Class 11 Maths Chapter 8 – Binomial Theorem

Quick revision notes

In Chapter 7 of Class 11 Maths: you must have learned about Permutations and Combinations. In chapter 8: Binomial Theorem of Class 11 Maths Chapter 8,  you will learn about the Binomial Theorem.

Binomial Theorem for Positive Integer

nC0nC1nC2, … , nno are called binomial coefficients and

nCr = n! / r!(n – r)! for 0 ≤ r ≤ n.

Properties of Binomial Theorem for Positive Integer

  1. Total number of terms in the expansion of (x + a)n is (n + 1).
  2. The sum of the indices of x and a in each term is n.
  3. The above expansion is also true when x and a are complex numbers.
  4. The coefficient of terms equidistant from the beginning and the end are equal. These coefficients are known as the binomial coefficients and

nCr = nCn  r, r = 0,1,2,…,n.

  1. General term in the expansion of (x + c)n is given by Tr + 1 = nCrxn – r ar.
  2. The values of the binomial coefficients steadily increase to maximum and then steadily decrease .

Middle term in the expansion of (1 + x)n

  1. It n is even, then in the expansion of (x + a)n, the middle term is (n/2 + 1)th terms.
  2. If n is odd, then in the expansion of (x + a)n, the middle terms are (n + 1) / 2 th term and (n + 3) / 2 th term.

Greatest Coefficient

  1. If n is even, then in (x + a)n, the greatest coefficient is nCn / 2
  2. Ifn is odd, then in (x + a)n, the greatest coefficient is nCn  1 / 2 or nCn + 1 / 2 both being equal.

Binomial Expression

An articulation comprising of two terms, associated by + or – sign is called binomial articulation. 

Binomial Theorem 

On the off chance that an and b are genuine numbers and n is a positive whole number, at that point 

The overall term of (r + 1)th term in the articulation is given by 

Tr+1 = nCr a r br 

Some Important Observations from the Binomial Theorem 

The absolute number of terms in the binomial expansion of (a + b)n is n + 1. 

The sum of the records of an and b in each term is n. 

The coefficient of terms equidistant from the earliest starting point and the end are equivalent. These coefficients are known as the binomial coefficient and 

nCr = nCn-r, r = 0, 1, 2, 3,… , n 

The values of the binomial coefficient consistently increment to a maximum and afterward consistently decline. 

The coefficient of xr in the expansion of (1 + x)n is nCr. 

In the binomial expansion (a + b)n, the rth term from the end is (n – r + 2)th term from the earliest starting point. 

Middle Term in the Expansion of series (a + b)n 

On the off chance that n is even, at that point in the expansion of (a + b)n, the middle term is

 (\frac { n }{ 2 } + 1) th term.

On the off chance that n is odd, at that point in the expansion of (a + b)n, the middle terms are (\frac { n+1 }{ 2 })th term and (\frac { n+3 }{ 2 })th term.

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