Pascal's triangle

Pascal's triangle is a triangular array of binomial coefficients in which each entry is the sum of the two entries directly above it. Its rows encode coefficients used in binomial expansions.

Binomial coefficient: A binomial coefficient counts selections of a fixed size from a finite set and appears as a coefficient in a binomial expansion. Each entry in row n is a binomial coefficient, generated from its two neighbors in the preceding row.
Combinations: Combinations are selections of objects in which order does not matter. Row entries count how many ways to select each possible number of objects from a fixed set.
Blaise Pascal: Blaise Pascal was a seventeenth-century French mathematician, physicist, and philosopher. His 1654 treatise systematized properties of the triangle, giving it its common Western name.
Fibonacci sequence: The Fibonacci sequence begins 0, 1, with each later term the sum of the two preceding terms. Adding shallow diagonals of the triangle produces Fibonacci numbers.
q-binomial coefficient: A q-binomial coefficient is a polynomial that generalizes an ordinary binomial coefficient using a parameter q. Its recurrence resembles Pascal's rule but introduces powers of q.
Binomial theorem: The binomial theorem gives the expansion of a power of a sum using binomial coefficients. Each row supplies the coefficients for expanding a power of a two-term sum.
Subset: A subset is a set whose elements all belong to another set. Summing a row counts all subsets of a finite set, grouped by their size.
Jia Xian: Jia Xian was an eleventh-century Chinese mathematician known for work on algebraic methods and root extraction. A triangular arrangement of binomial coefficients was associated with his method centuries before Pascal.
Triangular number: A triangular number counts objects arranged in an equilateral triangular pattern. The triangle's third diagonal lists triangular numbers.
Gaussian binomial coefficient: A Gaussian binomial coefficient counts subspaces of a finite-dimensional vector space over a finite field. It replaces ordinary subset counts with counts of vector subspaces and forms a q-analogue of the array.

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