binomial

A binomial is a polynomial with exactly two terms. Typical examples include a + b, x^2 + 3y, or 5uv. When a binomial is raised to a positive integer power n, its expansion is governed by the binomial theorem.

The binomial theorem states that for a nonnegative integer n, (x + y)^n equals the sum from k = 0 to n of binomial(n, k) x^{n-k} y^k, where binomial(n, k) = n! / (k! (n-k)!). The binomial coefficients binomial(n, k) form rows of Pascal's triangle and have a combinatorial interpretation as the number of ways to choose k items from a set of n. The theorem extends to real or complex exponents through the generalized binomial theorem, which yields an infinite series under suitable conditions.

In probability, the term binomial also describes the binomial distribution. A binomial distribution with parameters n and p models the number of successes in n independent Bernoulli trials each with success probability p. Its probability mass function is P(K = k) = binomial(n, k) p^k (1 - p)^{n - k} for k = 0, 1, ..., n. The distribution has mean np and variance np(1 - p).

Etymology and history: the word binomial comes from Latin binomium, meaning “two terms.” The concept of a binomial as a two-term polynomial has been central in algebra since the early modern period, with the binomial theorem later extended and generalized by mathematicians such as Isaac Newton.