binomialtheorie

Binomialtheorie, or the binomial theorem, describes the expansion of powers of a binomial expression. For any nonnegative integer n, (x+y)^n equals the sum from k=0 to n of binom(n,k) x^{n-k} y^k, where binom(n,k) = n!/(k!(n-k)!). These binomial coefficients appear in Pascal's triangle and count the number of ways to choose k items from n.

The identity can be proved by induction or by applying the distributive law to multiplying n copies of (x+y). The coefficients satisfy binom(n,k) = binom(n-1,k-1) + binom(n-1,k) and give a combinatorial interpretation as choosing k positions for y among n factors. The theorem provides a direct method to expand any power of a sum without multiplying out each term individually, and it underpins many counting and algebraic techniques.

Generalizations include Newton's binomial theorem for non-integer exponents: (1+x)^α = sum_{k=0}^∞ binom(α,k) x^k, with binom(α,k) = α(α-1)...(α-k+1)/k!, valid under certain convergence conditions (typically |x|<1). This extends the finite binomial theorem to real and complex exponents and leads to binomial series used in analysis. The binomial coefficients themselves also support various identities, recurrences, and connections to probability (the binomial distribution) and combinatorics, making binomialtheorie a central topic in algebra and discrete math.