Binomial Expansion Calculator

The binomial theorem states that (a + b)^n can be expanded as a sum of terms. Each term is of the form C(n,k) * a^(n-k) * b^k, where C(n,k) is the binomial coefficient 'n choose k'. The coefficients form Pascal's triangle, a pattern that appears throughout combinatorics and probability.

A binomial coefficient C(n,k) or 'n choose k' is the number of ways to choose k items from n items without regard to order. It is calculated as C(n,k) = n! / (k! * (n-k)!). For example, C(5,2) = 5! / (2! * 3!) = 10.

Pascal's triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The nth row contains the binomial coefficients for (a + b)^n. The triangle begins with 1, then 1,1, then 1,2,1, then 1,3,3,1, and so on. It has many remarkable properties and connections to probability, combinations, and algebra.

The expansion of (a + b)^n has exactly n + 1 terms. For example, (a + b)^3 has 4 terms, and (a + b)^5 has 6 terms. Each term has a different power of a and b, with the exponents of a and b always summing to n.

Binomial expansion is used in probability theory, combinatorics, statistical distributions (like the binomial distribution), approximations in calculus and physics, and computer science. It also appears in finance for option pricing models and in engineering for analyzing product reliability.