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$\begin{array}{cc}\text{Bernoulli Trials}& P\left(E\right)\text{Probability of event E: Get exactly k heads in n coin flips.}=\left(\genfrac{}{}{0}{}{n}{k}\right)\text{Number of ways to get exactly k heads in n coin flips}{p\text{Probability of getting heads in one flip}}_{}^{k\text{Number of heads}}{\left(1-p\right)\text{Probability of getting tails in one flip}}_{}^{n-k\text{Number of tails}}\\ \text{Cauchy-Schwarz Inequality}& {\left(\sum _{k=1}^{n}{a}_{k}^{}{b}_{k}^{}\right)}_{}^{2}\le \left(\sum _{k=1}^{n}{a}_{k}^{2}\right)\left(\sum _{k=1}^{n}{b}_{k}^{2}\right)\\ \text{Cauchy Formula}& f\left(z\right)\text{\hspace{0.17em}}·{\mathrm{Ind}}_{\gamma }^{}\left(z\right)=\frac{1}{2\pi i}\underset{\gamma }{\overset{}{\oint }}\frac{f\left(\xi \right)}{\xi -z}\text{\hspace{0.17em}}d\xi \\ \text{Cross Product}& {V}_{1}^{}×{V}_{2}^{}=|\begin{array}{ccc}i& j& k\\ \frac{\partial X}{\partial u}& \frac{\partial Y}{\partial u}& 0\\ \frac{\partial X}{\partial v}& \frac{\partial Y}{\partial v}& 0\end{array}|\\ \text{Vandermonde Determinant}& |\begin{array}{cccc}1& 1& \cdots & 1\\ {v}_{1}^{}& {v}_{2}^{}& \cdots & {v}_{n}^{}\\ {v}_{1}^{2}& {v}_{2}^{2}& \cdots & {v}_{n}^{2}\\ ⋮& ⋮& \ddots & ⋮\\ {v}_{1}^{n-1}& {v}_{2}^{n-1}& \cdots & {v}_{n}^{n-1}\end{array}|=\prod _{1\le i