Binomialformel

The binomial formula, also known as the binomial expansion, is a fundamental algebraic identity used to expand expressions of the form (a + b)^n. Here, 'a' and 'b' are terms, and 'n' is a non-negative integer exponent. The formula states that the expansion of (a + b)^n is a sum of terms, where each term is the product of a binomial coefficient, a power of 'a', and a power of 'b'.

The general form of the binomial formula is: (a + b)^n = sum from k=0 to n of (n choose k) a^(n-k) b^k. The term (n choose k) represents the binomial coefficient, which is calculated as n! / (k! (n-k)!), where '!' denotes the factorial. These coefficients can also be found in Pascal's triangle, where the nth row (starting from row 0) provides the coefficients for the expansion of (a + b)^n.

The powers of 'a' decrease from 'n' down to 0, while the powers of 'b' increase from 0 up to 'n'. The sum of the exponents in each term always equals 'n'. For example, the binomial formula for n=2 is (a + b)^2 = a^2 + 2ab + b^2. For n=3, it is (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3. The binomial formula is crucial in various fields, including algebra, calculus, probability theory, and combinatorics, for simplifying complex expressions and solving problems involving combinations and probabilities.