 Pascal's triangle

Row $n$ of Pascal's triangle consists of the binomial coefficients $\left(\genfrac{}{}{0ex}{}{n}{0}\right),\dots ,\left(\genfrac{}{}{0ex}{}{n}{n}\right)$, beginning with row $0$. Boundary entries are one, and Pascal's identity says each interior entry is the sum of the two entries above it. The hockey-stick identity evaluates sums along its diagonals.

The highlighted entry in the binomial coefficient illustration is $\left(\genfrac{}{}{0ex}{}{5}{2}\right)=10$. Split two-person teams according to whether they contain the fifth person: six choose both members from the other four, and four contain the fifth person together with one of those four. Thus $10=6+4$. All entries can be generated by the same local rule, so a large table of counts needs no repeated enumeration of its subsets.

Blaise Pascal's Traité du triangle arithmétique, published in 1665, presents an arithmetic array and connects it to combinations, games of chance, and powers of binomial expressions. This makes the connection between a counting table and several applications visible in the original work. ^[ref] The University of Cambridge's account of the treatise explains its 1654 composition and the organization of these applications. ^[ref]

Pascal's triangle is a triangular array of numbers that represents the binomial coefficients. Each number in the triangle is the sum of the two numbers directly above it in the previous row. The triangle starts with a single "1" at the top, known as the apex.