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} [...]