pascal triangle

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 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.