The probability function for the \(\BinomDistn(n, \pi)\) distribution can be found by counting how many distinct sequences of \(n\) Bernoulli trials that result in \(x\) successes, then multiplying this by the probability of any one such sequence.
Binomial probability function
If \(X\) has a \(\BinomDistn(n, \pi)\) distribution, its probability function is
\[ p(x)= {n \choose x} \pi^x(1-\pi)^{n-x} \qquad \text{for } x=0, 1, \dots, n \](Proved in full version)