Pascal's Triangle

The triangle begins with a single 1. The next rows are 1 1; 1 2 1; 1 3 3 1; and 1 4 6 4 1. Every row begins and ends with 1, and an interior entry is formed by adding the two entries immediately above it. In the common zero-based convention, the entry in position k of row n is the binomial coefficient n choose k, written C(n,k) = n!/[k!(n−k)!], for 0 ≤ k ≤ n. The same numbers may be displayed with the rows centered to make a triangle.

The entries in row n are the coefficients in the expansion of (x + y)^n. For example, row 4 gives (x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4. The binomial coefficient C(n,k) also counts the ways to choose k items from n distinct items when order does not matter. This links the triangle to combinations and to counting arguments in probability. Adding all entries in a row gives 2^n, because setting x and y to 1 in the binomial expansion counts all subsets of an n-item set.

Each row reads the same from left to right and right to left, reflecting the equality C(n,k) = C(n,n−k). The sums along the shallow diagonals form the Fibonacci sequence, while other diagonals contain familiar figurate-number sequences. These patterns arise from identities among binomial coefficients, not from the triangle's visual arrangement alone. Such relationships make the array a compact way to see connections among algebra, combinatorics and number sequences.

The familiar English name honors Blaise Pascal, whose study of the array appeared in a work published after his death in 1665. The naming does not mean that Pascal was the first person to know the pattern. Encyclopaedia Britannica describes an 11th-century triangular arrangement of binomial coefficients devised by the Chinese mathematician Jia Xian and its later study by Yang Hui; Persian scholar Omar Khayyam also studied the pattern in the 11th century. Yang Hui's 13th-century mathematical writing includes the triangle, and it is often called Yang Hui's triangle in China. Niccolò Tartaglia published rows of it in Europe in the 16th century. These records show that the array was developed and used across different mathematical traditions before Pascal's treatment.