Seminar in Mathematics: Binomial Theorem

In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power (x + y)n into a sum involving terms of the form ax^by^c, where the coefficient of each term is a positive integer, and the sum of the exponents of x and y in each term is n. For example,

The coefficients appearing in the binomial expansion are known as binomial coefficients. They are the same as the entries of Pascal's triangle, and can be determined by a simple formula involving factorials. These numbers also arise in combinatorics, where the coefficient of x^n−ky^k is equal to the number of different combinations of k elements that can be chosen from an n-element set.

The binomial coefficients appear as the entries of Pascal's triangle.

Statement of the Theorem

According to the theorem, it is possible to expand any power of x + y into a sum of the form

where

denotes the corresponding binomial coefficient. Using summation notation, the formula above can be written

This formula is sometimes referred to as the Binomial Formula or the Binomial Identity.

A variant of the binomial formula is obtained by substituting 1 for x and x for y, so that it involves only a single variable. In this form, the formula reads

or equivalently

Examples
The most basic example of the binomial theorem is the formula for square of x + y:

The binomial coefficients 1, 2, 1 appearing in this expansion correspond to the third row of Pascal's triangle. The coefficients of higher powers of x + y correspond to later rows of the triangle:

The binomial theorem can be applied to the powers of any binomial. For example,

For a binomial involving subtraction, the theorem can be applied as long as the negation of the second term is used. This has the effect of negating every other term of the expansion: