General Binomial Theorem

General Binomial Theorem

\( \begin{align} \displaystyle (a+b)^n &= \binom{n}{0}a^nb^0 + \binom{n}{1}a^{n-1}b^1 + \cdots + \binom{n}{k}a^{n-k}b^{k} + \cdots + \binom{n}{n}a^{0}b^{n} \\ &= \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^{k} \\ \end{align} \) \( \begin{align} \displaystyle 1^{\text{st}} \text{ term } T_1 &= \binom{n}{0}a^nb^0 \\ 2^{\text{nd}} \text{ term } T_2 &= \binom{n}{1}a^{n-1}b^1 \\ 3^{\text{rd}} \text{ term } T_3 &= \binom{n}{2}a^{n-2}b^2 \\ &\vdots \\ k^{\text{th}} \text{ term } […]

Binomial Coefficient

Binomial Coefficient

$$\binom{n}{k}=\dfrac{n!}{k!(n-k)!}$$Note that the binomial coefficient is sometimes written $^nC_k$ or $C^n_k$, depending on authors or geographical regions. \( \begin{aligned}\binom{n}{k} &= \dfrac{n!}{k!(n-k)!} \cdots (1) \\\binom{n}{n-k} &= \dfrac{n!}{(n-k)!(n-(n-k))!} = \dfrac{n!}{(n-k)!k!} \cdots (2) \\\therefore \binom{n}{k} &= \binom{n}{n-k} \text{by } (1) \text{ and } (2) \\\end{aligned} \) This means;\( \begin{aligned}\binom{10}{2} &= \binom{10}{8} \\\binom{100}{1} &= \binom{100}{99} \\\end{aligned} \) The following […]

Binomial Expansion | Binomial Theorem

Binomial Expansion | Binomial Theorem

Binomial Expansion is based on two terms, that is binomial.Any expression of the form \( (a+b)^n \) is called the power of a binomial.All binomials raised to a power can be expanded using the same general principles. \( \begin{aligned} \displaystyle(a+b)^1 &= a+b \\(a+b)^2 &= (a+b)(a+b) \\&= a^2+2ab+b^2 \\(a+b)^3 &= (a+b)(a^2+2ab+b^2) \\&= a^3+3a^2b+3ab^2+b^3 \\(a+b)^4 &= (a+b)(a^3+3a^2b+3ab^2+b^3) […]