# Number Theory: Binomial theorem

## Understanding the Binomial Theorem

• The Binomial Theorem describes the algebraic expansion of powers of a binomial. The theorem states that for any integers n and k, it is possible to expand the power (a + b)^n into a sum involving terms of the form ax^ky^n-k.
• According to the binomial theorem, the coefficient of x^ky^n-k in the expansion of (x + y)^n is given by nCk, where C represents the combination operator, also known as the binomial coefficient.
• The binomial theorem implies that there are “n+1” terms in the expansion (a + b)^n.

## Applying the Binomial Theorem

• The binomial coefficients in the expansion follow a specific pattern, corresponding to the entries in Pascal’s Triangle. Each row in Pascal’s triangle gives the coefficients for the expansion of a binomial to the power of the row number, starting from the 0th row.
• The general term in the expansion of (a+b)^n is given by T_(r+1) = ^nC_r * a^(n-r) * b^r where r varies from 0 to n.

## The Binomial Series

• The binomial theorem extends to rational and real number exponents, resulting in the binomial series.
• When expanding (a+b)^n, if n is not a positive integer, then the binomial series becomes an infinite series.
•  The binomial series denotes the expanded form of (1 + x)^n where x < 1 and n ε R.

## Solving Problems using Binomial Theorem

• The binomial theorem proves beneficial while solving problems involving combinatorics, algebra, and calculus.
• A crucial application of the binomial theorem is in calculations involving powers of numbers. The theorem helps break down the calculation into smaller, manageable parts.

## Other Facts about the Binomial Theorem

• The binomial theorem was first proved by mathematician Newton, but it already was in use centuries before by ancient mathematicians.
• Binomial theorem forms the basis for the theory of binomial distributions in statistics.