Binomial coefficient

The binomial coefficient \(\binom{n}{k}\) counts the ways to choose \(k\) elements from an \(n\)-element set and is the coefficient of \(x^k y^{n-k}\) in \((x+y)^n\).

Pascal's identity: The recurrence \(\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}\). Splitting choices by whether they include a fixed element gives this recurrence.
Binomial distribution: The probability distribution for the number of successes in a fixed number of independent trials with equal success probability. Its probability formula weights each success count by the corresponding binomial coefficient.
Combination: A selection of objects in which order does not matter. A binomial coefficient counts combinations of a fixed size.
Permutation: An ordered arrangement of distinct objects. Unlike a binomial coefficient, permutations distinguish different orders of the same selection.
Factorial: The product of all positive integers up to a given integer, with \(0!=1\). The formula \(\binom{n}{k}=n!/(k!(n-k)!)\) computes the count.
Bernoulli trial: A random experiment with two outcomes, usually called success and failure. A sequence of these trials has \(\binom{n}{k}\) outcome patterns with exactly \(k\) successes.
Combinatorial proof: A proof of an identity by showing that both sides count the same objects. Many binomial identities become clear by counting one collection in two ways.
Multinomial coefficient: A coefficient counting partitions of a set into several specified size classes. It generalizes the two-class choice counted by a binomial coefficient.
Pascal's triangle: A triangular arrangement of numbers in which each interior entry is the sum of the two above it. Its entries list binomial coefficients row by row.
Finite difference: The change between values of a sequence at successive integer inputs. Binomial coefficients provide coefficients for repeated finite-difference formulas.

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