The user-friendly version of this content is available here.
The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.
This essential trains for: SAT-II, AMC-10, AMC-12, AIME.
The binomial theorem is a recipe for expanding the power of a binomial expression (a binomial is an expression with two unlike terms):
Assuming that n is a positive integer, here are some simple examples:
We can derive a formula for an unspecified exponent by thinking how the terms of the resulting polynomial are produced: by collecting like terms after the multiplication of all n identical binomial factors.
The term xy appears two times and results with a coefficient of 2 after collecting like terms.
For the next power:
if we look, for example, at how the term x2y is produced, we see that it is formed by:
In other words, if the degree is 3 we are looking for a "string of three letters" such that two of them are "x" and one of them is "y." According to what we know about linear permutations with repetitions, the number of such strings is:
where from we get the coefficient of 3. This is called a binomial coefficient. It is also called a combination or a choose.
Any other binomial coefficient is derived in the same way. For example in:
Let us derive the coefficient for the term that contains xk. Consider in how many ways you can place x k times in n slots in order to find the coefficient of the term that contains xk. Because the degree of each term is n, the exponent of y can only be n - k.
Therefore, the term is:
Since by choosing positions for x in the 'string', the positions for the y symbols are already chosen, the binomial coefficients must be symmetric, i.e. the coefficient of xkyn-k must be equal to the coefficient of xn-kyk.
The entire expansion can be written:
Of course, the coefficients of the first and last terms can also be written as binomial coefficients, we simply did not bother since:
One of the notations for binomial coefficients is:
Using the summation symbol, the binomial expansion can be written as:
Example: Find the ninth term of the expansion:
The ninth term is obtained from the binomial formula with k = 8 (since k takes values starting at zero!):