Expanding (a+b)² by hand is easy: a² + 2ab + b². Expanding (a+b)¹⁰ by repeated multiplication is a genuine slog — eleven terms, each with its own coefficient, and getting there by multiplying (a+b) by itself ten times in a row invites errors at every step. The binomial theorem skips all of that multiplication entirely, handing over the exact coefficient of every single term directly, calculated from a simple, well-known pattern.
The coefficients were already sitting in a triangle
The binomial theorem states that the coefficients in the expansion of (a+b)ⁿ are exactly the numbers in the nth row of what's popularly known as Pascal's Triangle — a triangular arrangement where each number is the sum of the two numbers diagonally above it, starting from a single 1 at the top. The row for n=2 reads 1, 2, 1 — matching a² + 2ab + b² exactly. The row for n=10 gives all eleven coefficients for (a+b)¹⁰ directly, without a single multiplication of a or b required to find them. Each entry in the triangle is also expressible as a combination — the number of ways to choose k items from a set of n — connecting the expansion directly to combinatorics, the branch of mathematics concerned with counting.
Named for Pascal, known long before him
Despite the common Western name, the triangular pattern of coefficients was documented centuries before Blaise Pascal's 17th-century work on it — Indian, Persian, and Chinese mathematicians had independently described the same triangle and its connection to binomial expansion, in some cases many hundreds of years earlier. Pascal's specific contribution was a more systematic treatise connecting the triangle to probability theory, and it's largely through that later, widely circulated work that the triangle picked up his name in much of the West — a naming pattern common across mathematical history, where the person most associated with popularising or systematising a result in a particular tradition often gets remembered over earlier independent discoverers working in other traditions.
What we're still unsure about
The binomial theorem itself, and its connection to Pascal's Triangle and combinatorics, is fully proven, elementary mathematics — there's no live dispute about its correctness. What's more genuinely a matter of historical record-keeping than mathematics is establishing precisely who first discovered which specific results, and how independently different mathematical traditions arrived at closely related versions of the pattern — Indian mathematician Pingala's work dates back over two thousand years and touches on related combinatorial ideas, and historians of mathematics continue refining exactly how much direct or indirect influence passed between traditions versus how much was genuinely independent rediscovery of a pattern that, in some sense, was always there to be found.
This sits inside Binomial Theorem, one of eight topics in Algebra, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.