binom newton

It seems you might be referring to the Binomial Theorem, which is often associated with Sir Isaac Newton. The Binomial Theorem provides a formula for expanding expressions of the form \((a + b)^n\), where \(n\) is a non-negative integer. The theorem states that: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] In this equation: - \( \binom{n}{k} \) (read as "n choose k") is the binomial coefficient, calculated as: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] - \( a \) and \( b \) are any numbers (or variables). - \( n \) is a non-negative integer. - The sum is taken from \( k = 0 \) to \( n \). ### Example of the Binomial Theorem For example, to expand \((x + y)^3\): \[ (x + y)^3 = \sum_{k=0}^{3} \binom{3}{k} x^{3-k} y^k \] Calculating this gives: \[ = \binom{3}{0} x^3 y^0 + \binom{3}{1} x^2 y^1 + \binom{3}{2} x^1 y^2 + \binom{3}{3} x^0 y^3 \] \[ = 1 \cdot x^3 + 3 \cdot x^2y + 3 \cdot xy^2 + 1 \cdot y^3 \] \[ = x^3 + 3x^2y + 3xy^2 + y^3 \] If you are looking for something more specific regarding Newton and binomials or need further elaboration, feel free to ask! Update (2024-08-22): It seems like you're referring to the binomial theorem, which is often associated with the mathematician Isaac Newton. The binomial theorem provides a formula for expanding expressions that are raised to a power, specifically for binomials (expressions that have two terms). The binomial theorem states that: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] where: - \( n \) is a non-negative integer, - \( a \) and \( b \) are any numbers, - \( \binom{n}{k} \) (called a binomial coefficient) is calculated as \( \frac{n!}{k!(n-k)!} \), - \( k \) is an integer ranging from 0 to \( n \). The binomial coefficients \( \binom{n}{k} \) represent the number of ways to choose \( k \) successes in \( n \) trials and are commonly represented in Pascal's Triangle. For example, if you expand \( (x + y)^2 \), you would get: \[ (x + y)^2 = \binom{2}{0} x^2 y^0 + \binom{2}{1} x^1 y^1 + \binom{2}{2} x^0 y^2 = x^2 + 2xy + y^2 \] If you have a specific question or need clarification on a particular aspect of the binomial theorem, feel free to ask!