Binomial Coefficient

The binomial coefficient $\binom{n}{k}$ (read “$n$ choose $k$”) counts the number of combinations of $k$ elements from a set of $n$. It is defined as $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ for $0 \leq k \leq n$.

Binomial coefficients satisfy the recurrence $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$, which generates Pascal’s triangle. They appear as coefficients in the binomial theorem: $(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}$.

The symmetry $\binom{n}{k} = \binom{n}{n-k}$ reflects the fact that choosing $k$ elements to include is the same as choosing $n - k$ elements to exclude.