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 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) […]