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.

## Factorial Notation

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

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.

## Binomial Expansion

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:

where

(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.

### Example 4

Find the first three terms, in descending powers of x, of the binomial expansion of .

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

### Sample Exam Question

- Given that , write down the value of q.
- Given that the coefficient of in the expansion of is 28. find the value of p
- Using the first three terms of a binomial expansion, estimate the value of .

*Answers:*

- The formula for ‘n choose r’ is given by . Setting n=8 and r=2 gives the missing term n-r=6 and so q=6.
- One can perform the full expansion up to the term or notice that only the coefficient of is required. That is, the coefficient when the term is simplified. Notice that a and b are interchangeable in the formula and are chosen so that the power of x is 2. The coefficient is thus and equal to 28 giving p=2.
- Find the first 3 terms in the binomial expansion using p=2: Note that we already knew the coefficient of the term. Using this expansion suggests that we should choose x so that , that is, or x=0.04. Substituting x=0.04 into the expansion gives The actual answer to 7 decimal places, using a calculator, is 250.9245767, so not a great approximation.

## Statistics Example

Consider the binomially distributed random variable . Find the probability in terms of x. Write your answer as a polynomial in x.

Using the formula:

We can use Example 2 above to expand :

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