binom6

Binom6 is a term used to denote the binomial coefficients for n = 6, i.e., the sixth row of Pascal's triangle. The values are C(6, k) for k = 0, 1, ..., 6, which give the sequence 1, 6, 15, 20, 15, 6, 1. By definition C(6, k) = 6! / (k! (6−k)!). In the binomial theorem, these coefficients appear in the expansion of (x + y)^6: x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6.

Combinatorial interpretation: C(6,k) is the number of ways to choose k elements from a 6-element set. The row is symmetric, reflecting C(6,k) = C(6,6−k). The row 6 is the sixth line in Pascal's triangle.

Applications include counting problems in probability and combinatorics, and evaluating binomial distributions with n = 6 where the probability of exactly k successes is C(6,k) p^k (1−p)^(6−k).

Although not a universally standardized term, binom6 is sometimes used informally to refer to this fixed-n collection of binomial coefficients or the sixth row of Pascal's triangle.