Binomial coefficient

For a binomial variable $X$, the number of possible combinations of $x$ successes in $n$ trials equals $$\begin{equation*}{n\choose x}=\frac{n!}{x!(n-x)!}, \end{equation*}$$ where $0\le x\le n$ and $n!$ ($n$-faculty) $=n(n-1)(n-2)\ldots 1$. By definition: $0!=1$.

Formula binomial distribution

For a binomial variable $X$, the probability of $x$ successes in $n$ trials is $$\begin{equation*} P(X=x)={n\choose x}p^x(1-p)^{n-x}, \end{equation*}$$ where $p$ is the probability of success.

Expected value and standard deviation

The expected value of a discrete random variabele $X$ with $k$ possible outcomes, equals the mean of the distribution $$\begin{equation*} \mu= \sum_{i=1}^k{x_iP(x_i)}. \end{equation*}$$ The standard deviation of this random variable is $$\begin{equation*}\sigma= \sqrt{\sum_{i=1}^k{(x_i -\mu)^2P(x_i)}}.\end{equation*}$$ For a dichotomous (binary) binomial variable simpler formulas are available. The expected value of probability of success $p$ with $n$ trials is $$\begin{equation*} \mu= np \end{equation*}$$ and the standard deviation is $$\begin{equation*} \sigma= \sqrt{np(1-p)}. \end{equation*}$$