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)ⁿ = ⁿC_oxⁿ + ⁿC₁x^n-1a + ⁿC₂x^n-2a² + ⁿC₃x^n-3a³ + …. + ⁿC_naⁿ

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

(x – a)ⁿ = ⁿC_oxⁿ – ⁿC₁x^n-1a + ⁿC₂x^n-2a² – ⁿC₃x^n-3a³ + …. – ⁿC_naⁿ

where, x and a are real numbers and ⁿC_o, ⁿC₁, ⁿC₂, …,  ⁿC_n 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

T_r+1 = ⁿC_rx^n-ra^r

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

T_r+1 = (-1)^r nC_rx^n-ra^r

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 C_o, C₁, C₂, …, C_n are the coefficients of (1 + x)ⁿ, then

(i) ⁿC_r + ⁿC_{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) ⁿC₀ + ⁿC₁ + ⁿC₂ + ⁿC₃ + … + ⁿC_n = 2ⁿ

(v) ⁿC₀ + ⁿC₂ + ⁿC₄ + … = ⁿC₁ + ⁿC₃ + ⁿC₅ + … = 2^n-1