Binomial Expansion refers to expanding an expression that involves two terms added together and raised to a power, i.e. (x+y)^n. 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:

6!=6\times 5\times 4 \times 3 \times 2 \times 1=720

Here are some more examples:

8!=8\times 7\times 6\times 5\times 4 \times 3 \times 2 \times 1=40,320

4!\times 3!=4\times 3\times 2 \times 1\times 3\times 2 \times 1=24\times 6=144

9!\div 5!=\frac{9\times 8\times 7\times 6\times 5\times 4 \times 3 \times 2 \times 1}{5\times 4 \times 3 \times 2 \times 1}=9\times 8\times 7\times 6=3,024


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 ^{6}C_{2}, pronounced ‘6 choose 2’. This can also be written as {6}\choose{2}. In general, we write ^{n}C_r or {n}\choose {r} and is calculated as

^{n}C_r=\left(\begin{array}{c}n\\r\end{array}\right)=\frac{n!}{(n-r)!r!}

This comes from summing all the terms above a given entry and simplifies to a fraction with factorials. ^{n}C_r 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 x^ry^{n-r} term in the expansion of (x+y)^n. Hence, this is why Pascal’s triangle is useful in Binomial Expansion. Note that there is a button on your calculator for working out {n}\choose{r} – you don’t necessarily need to calculate the individual factorials. You might also notice that {{n}\choose{0}}={{n}\choose{n}}=1 and {{n}\choose{1}}={{n}\choose{n-1}}=n always.


Binomial Expansion

Suppose now that we wish to expand (x+y)^n, 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.

Example 1

Expand (x+y)^3.


Example 2

Using Example 1 expand (x+y)^4.


Expanding (x+y)^n by hand for larger n becomes a tedious task. The Edexcel Formula Booklet provides the following formula for binomial expansion:

(a+b)^n=a^n+\left(\begin{array}{c}n\\1\end{array}\right)a^{n-1}b+\left(\begin{array}{c}n\\2\end{array}\right)a^{n-2}b^2+...+\left(\begin{array}{c}n\\r\end{array}\right)a^{n-r}b^r+...+b^n

where

\left(\begin{array}{c}n\\r\end{array}\right)=\frac{n!}{(n-r)!r!}

(see above) for when n\in{\mathbb N}, 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 3

Find the first three terms in the expansion (x+y)^8.


Example 4

Find the first three terms, in descending powers of x, of the binomial expansion of (2x+4)^5.


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 p^r) we require the remaining n-r trials to be unsuccessful (probability (1-p)^{n-r}). The number of combinations in which there can be r successes out of n trials is {n}\choose{r} (see above). Finally, the associated probability is given by

P(X=r)={{n}\choose{r}}p^r(1-p)^{n-r}

when X\sim B(n,p)  as seen on the Binomial Distribution page.


More Binomial Expansion Examples


Sample Exam Question

  1. Given that {{8}\choose{2}}=\frac{8!}{2!q!}, write down the value of q.
  2. Given that the coefficient of x^2 in the expansion of \left(p-\frac{x}{8}\right)^8 is 28. find the value of p
  3. Using the first three terms of a binomial expansion, estimate the value of 1.995^8.

Answers:

  1. The formula for ‘n choose r’ is given by {{n}\choose{r}}=\frac{n!}{r!(n-r)!}. Setting n=8 and r=2 gives the missing term n-r=6 and so q=6.
  2. One can perform the full expansion up to the x^2 term or notice that only the coefficient of x^2 is required. That is, the coefficient when the term {{8}\choose{2}}p^6\left(-\frac{x}{8}\right)^2 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 28\times p^6\times \frac{1}{64} and equal to 28 giving p=2.
  3. Find the first 3 terms in the binomial expansion using p=2:\left(2-\frac{x}{8}\right)^8\approx 2^8-{{8}\choose{1}}\times 2^7\times-\frac{x}{4}+28 x^2=256-256x+28x^2 Note that we already knew the coefficient of the x^2 term. Using this expansion suggests that we should choose x so that 2-\frac{x}{8}=1.995, that is, 0.005=\frac{x}{8} or x=0.04. Substituting x=0.04 into the expansion gives (1.995)^8\approx256-256\times 0.04+28\times 0.04^2=245.8048  The actual answer to 7 decimal places, using a calculator, is 250.9245767, so not a great approximation.


Binomial Expansion Board Question

binomial expansion

Note that C2 was a module when A-Level Maths was delivered in a modular fashion. Binomial Expansion is now an AS-Level topic.

Statistics Example

Consider the binomially distributed random variable X\sim B(7,x). Find the probability P(X=3) in terms of x. Write your answer as a polynomial in x.

Using the formula:

P(X=3)={{7}\choose{3}}x^3(1-x)^4

We can use Example 2 above to expand (1-x)^4:

P(X=3)=35x^3\left(1-4x+6x^2-4x^3+x^4\right)

=35x^3-140x^4+210x^5-140x^4+35x^3

Click here to find Questions by Topic and scroll down to all past BINOMIAL EXPANSION exam questions to practice some more.


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