Binomial Theorem

The binomial theorem provides a simple method for determining the coefficients of each term in the expansion of a binomial with the general equation (A + B)ⁿ. Developed by Isaac Newton, this theorem has been used extensively in the areas of probability and statistics. The main argument in this theorem is the use of the combination formula to calculate the desired coefficients.

The question of expanding an equation with two unknown variables called a binomial was posed early in the history of mathematics. One solution, known as Pascal's triangle, was determined in China as early as the thirteenth century by the mathematician Yang Hui. His solution was independently discovered in Europe 300 years later by Blaise Pascal whose name has been permanently associated with it since. The binomial theorem, a simpler and more efficient solution to the problem, was first suggested by Isaac Newton. He developed the theorem as an undergraduate at Cambridge and first published it in a letter written for Gottfried Leibniz, a German mathematician.

Expanding an equation like (A + B)ⁿ just means multiplying it out. By using standard algebra the equation (A + B)² can be expanded into the form A² + 2AB + B². Similarly, (A + B)⁴can be written A⁴ + 4A³B + 6A² B² + 4AB³ + B⁴. Notice that the terms for A and B follow the general pattern AⁿB⁰,A^n-1B¹,A^n-2B²,A^n-3B³,...,A¹B^n-1, A⁰Bⁿ. Also observe that as the value of n increases, the number of terms increases. This makes finding the coefficients for individual terms in an equation with a large n value tedious. For instance, it would be cumbersome to find the coefficient for the term A⁴B³ in the expansion of (A + B)⁷ if we used this algebraic approach. The inconvenience of this method led to the development of other solutions for the problem of expanding a binomial.

One solution, known as Pascal's triangle, uses an array of numbers (shown below) to determine the coefficients of each term.

This triangle of numbers is created by following a simple rule of addition. Numbers in one row are equal to the sum of two numbers in the row directly above it. In the fifth row the second term, 4 is equal to the sum of the two numbers above it, namely 3 + 1. Each row represents the terms for the expansion of the binomial on the left. For example, the terms for (A+B)³ are A³ + 3A²B + 3AB² + B³. Obviously, the coefficient for the terms A³ and B ³ is 1. Pascal's triangle works more efficiently than the algebraic approach, however, it also becomes tedious to create this triangle for binomials with a large n value.

The binomial theorem provides an easier and more efficient method for expanding binomials which have large n values. Using this theorem the coefficients for each term are found with the combination formula. The combination formula is

The notation n! is read "n factorial" and means multiplying n by every positive whole integer which is smaller than it. So, 4! would be equal to 4 × 3 × 2 × 1 = 24. Applying the combination formula to a binomial expansion (A + B)ⁿ, n represents the power to which the formula is expanded, and r represents the power of B in each term. For example, for the term A⁴B³ in the expansion of (A + B)⁷, n is equal to 7 and r is equal to 3. By substituting these values into the combination formula we get 7! / (3! × 4!) = 35, which is the coefficient for this term. The complete binomial theorem can be stated as the following: