Table of Contents

Toggle**Binomial Expression**

An algebraic expression consisting of two terms with a positive and negative sign between them is called binomial expression.

## Binomial Theorem

If n is any positive integer, then

(x + a)^{n} = ^{n}C_{o}x^{n} + ^{n}C_{1}x^{n}^{-1}a + ^{n}C_{2}x^{n}^{-2}a^{2} + ^{n}C_{3}x^{n}^{-3}a^{3} + …. + ^{n}C_{n}a^{n}

i.e. (x + a)^{n} = \sum_{i=1}^{n}{^nC_rx^{n-r}a^r} — Binomial Theorem

(x – a)^{n} = ^{n}C_{o}x^{n} – ^{n}C_{1}x^{n}^{-1}a + ^{n}C_{2}x^{n}^{-2}a^{2} – ^{n}C_{3}x^{n}^{-3}a^{3} + …. – ^{n}C_{n}a^{n}

where, x and a are real numbers and ^{n}C_{o}, ^{n}C_{1}, ^{n}C_{2}, …, ^{n}C_{n} are called binomial coefficients.

Also, n!\over r! (n - r)! 0≤ r ≤ n

## Properties of Binomial Theorem for Positive Integer

- The total number of terms in the expression of (x + a)
^{n}is (n + 1). - The sum of the indices of x and an in each term in n.
- The above expansion is also true when x and a are complex numbers.
- The values of the binomial coefficients steadily increase to a maximum and then steadily decrease.

## General Term in a Binomial Expansion

(i) General term in the expansion of (x + a)^{n} is

T_{r+1} = ^{n}C_{r}x^{n-r}a^{r}

(ii) General term in the expansion of (x – a)^{n} is

T_{r+1} = (-1)^{r} ^{n}C_{r}x^{n-r}a^{r}

## Middle Term in a Binomial Expansion

- If n is even in the expansion of (x + a)
^{n}or (x – a)^{n}, then the middle term is \bigg({n\over2}+1\bigg) th term. - If n is odd in the expansion of (x + a)
^{n}or (x – a)^{n}, then the middle term is \bigg({n+1\over2}\bigg) th term and \bigg({n+1\over2}+1\bigg) th term

**Note: **Where there are two middle terms in the expansion, then their binomial coefficients are equal.

## Important Results on Binomial Coefficient

If C_{o}, C_{1}, C_{2}, …, C_{n} are the coefficients of (1 + x)^{n}, then

(i) ^{n}C_{r} + ^{n}C_{r – 1} = ^{n + 1}C r

(ii) { {^n}C_r \over{^{n-1}}C_{r-1}}={n\over r}

(iii) { {^n}C_r \over{^{n}}C_{r-1}}={n-r+1\over r}

(iv)^{ n}C_{0} + ^{n}C_{1} + ^{n}C_{2} + ^{n}C_{3} + … + ^{n}C_{n} = 2^{n}

(v) ^{n}C_{0} + ^{n}C_{2} + ^{n}C_{4} + … = ^{n}C_{1} + ^{n}C_{3} + ^{n}C_{5} + … = 2^{n-1}