binomial coefficient

The factorial of a non-negative integer \(n\), denoted by \(n!\), is the product of all positive integers from 1 to \(n\).

\[ n! = \prod_{k=1}^{n} k \]

The binomial coefficient represents the number of distinct ways to choose \(k\) elements from a larger set of \(n\) elements, where the order of selection does not matter. The formula for the binomial coefficient is:

\[
\binom{n}{k} = \begin{cases}
\displaystyle\frac{n!}{k!\\,(n-k)!} & \text{if } 0 \leq k \leq n \\[1em]
0 & \text{if } k > n
\end{cases}
\]

The inclusion-exclusion principle is a counting rule that determines the size of a union of finite sets by accounting for their overlaps.

The theorem states that for any given positive integer \(n\), the expansion of the binomial expression \((a+b)^n\) can be expressed as the sum of \(n+1\) terms, where each term is a coefficient multiplied by the product of the two binomial expressions \(a\) and \(b\), each raised to a power.

The binomial distribution models the probability of obtaining a given number of successes in independent Bernoulli trials with constant probability p. Formally, the binomial distribution is expressed as

\[
b(x; n, p) = \binom{n}{x} p^{x} q^{n – x}
\]