Binomial Expansion refers to expanding an expression that involves two terms added together and raised to a power, i.e. . Before learning how to perform a Binomial Expansion, one must understand factorial notation and be familiar with Pascal’s triangle.
When you see an exclamation mark following a number in mathematics it is known as a factorial. For example, 6! is said ‘6 factorial’ and you multiply all of the positive integers less than 6 together:
Here are some more examples:
Pascal’s triangle is the pyramid of numbers where each row is formed by adding together the two numbers that are directly above it:
0th row: 1
1st row: 1 1
2nd row: 1 2 1
3rd row: 1 3 3 1
4th row: 1 4 6 4 1
5th row: 1 5 10 10 5 1
6th row: 1 6 15 20 15 6 1
7th row: 1 7 21 35 35 21 7 1
The triangle continues on this way, is named after a French mathematician named Blaise Pascal (find out more about Blaise Pascal) and is helpful when performing Binomial Expansions.
Notice that the 5th row, for example, has 6 entries. Like the 0th row, the first entry in any one row is the 0th entry. Consider the first 15 in the 6th row, we call this , pronounced ‘6 choose 2’. This can also be written as . In general, we write or and is calculated as
This comes from summing all the terms above a given entry and simplifies to a fraction with factorials. can be thought of as the number of combinations of putting r balls in n buckets. It is also the number of times you get an term in the expansion of . Hence, this is why Pascal’s triangle is useful in Binomial Expansion. Note that there is a button on your calculator for working out – you don’t necessarily need to calculate the individual factorials. You might also notice that and always.
Suppose now that we wish to expand , i.e. find the Binomial Expansion. In the simple case where n is a relatively small integer value, the expression can be expanded one bracket at a time. See Examples 1 and 2 below.
Expanding by hand for larger n becomes a tedious task. The Edexcel Formula Booklet provides the following formula for binomial expansion:
(see above) for when , i.e for when n is a positive integer. Directly substituting x in place of a and y in place of b results in finding the expansions for larger n. Usually only the first few terms are required – see Example 3. You may substitute other expressions or numbers for a and b and you may be asked for ascending or descending powers of a particular variable. See Example 4 – you will notice that when there are also coefficients inside the brackets, the coefficients in the expansion change dramatically from those given in Pascal’s triangle.
Now check out the further examples below to see what an exam question might look like.
Relationship to Probabilities
Consider a binomially distributed random variable with n trials and probability of success p – see Binomial Distribution. If we require r of the trials to be successful (probability ) we require the remaining n-r trials to be unsuccessful (probability ). The number of combinations in which there can be r successes out of n trials is (see above). Finally, the associated probability is given by
when as seen on the Binomial Distribution page.
More Binomial Expansion Examples
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