How does Pascal’s triangle help expand binomial expressions?

Pascal’s triangle is a triangular arrangement of numbers that provides the binomial coefficients used to expand binomial expressions of the form $(x+y{)}^n$.1

The triangle is constructed by placing a 1 at the start and end of each row, with each interior number being the sum of the two numbers directly above it. To expand a binomial expression, you identify the row in the triangle that corresponds to the exponent $n$. The numbers in that row serve as the coefficients for the terms in the expansion.123

For example:
* Row 0 (1): $(x+y{)}^0=1$
* Row 1 (1 1): $(x+y{)}^1=x+y$
* Row 2 (1 2 1): $(x+y{)}^2=x^2+2xy+y^2$
* Row 3 (1 3 3 1): $(x+y{)}^3=x^3+3x^{2}y+3x{y}^2+y^3$

While named after the 17th-century French mathematician Blaise Pascal, the array was known to earlier mathematicians, including Al-Karajī and Jia Xian, who utilized it to solve algebraic problems centuries prior.123