Question

state and prove binomial theorem​

Answer 1

Answer:

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (x + y)n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. For example (for n = 4),

{\displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.}{\displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.}

The coefficient a in the term of axbyc is known as the binomial coefficient {\displaystyle {\tbinom {n}{b}}}{\displaystyle {\tbinom {n}{b}}} or {\displaystyle {\tbinom {n}{c}}}{\displaystyle {\tbinom {n}{c}}} (the two have the same value). These coefficients for varying n and b can be arranged to form Pascal's triangle. These numbers also arise in combinatorics, where {\displaystyle {\tbinom {n}{b}}}{\displaystyle {\tbinom {n}{b}}} gives the number of different combinations of b elements that can be chosen from an n-element set. Therefore {\displaystyle {\tbinom {n}{b}}}{\displaystyle {\tbinom {n}{b}}} is often pronounced as "n choose b".