# Binomial Theorem Formula

## 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)nnCoxn + nC1xn-1a + nC2xn-2a2 + nC3xn-3a3 + …. + nCnan

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

(x – a)nnCoxnnC1xn-1a + nC2xn-2a2nC3xn-3a3 + …. – nCnan

where, x and a are real numbers and nCo, nC1, nC2, …,  nCn are called binomial coefficients.

Also, n!\over r! (n - r)! 0≤ r ≤ n

## Properties of Binomial Theorem for Positive Integer

1. The total number of terms in the expression of (x + a)n is (n + 1).
2. The sum of the indices of x and an in each term in n.
3. The above expansion is also true when x and a are complex numbers.
4. 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

Tr+1nCrxn-rar

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

Tr+1 = (-1)r nCrxn-rar

## Middle Term in a Binomial Expansion

1. 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.
2. 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 Co, C1, C2, …, Cn are the coefficients of (1 + x)n, then

(i) nCrnCr – 1n + 1C 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) nC0nC1nC2nC3 + … + nCn = 2n

(v) nC0nC2nC4 + … = nC1nC3nC5 + … = 2n-1

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