Binomial Expansion – positive integer powers
Binomial Expansion Notes
Binomial Expansion is essentially multiplying out brackets. A binomial is two terms added together and this is 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:
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, we expand the expression one bracket at a time. See Examples 1 and 2. Expanding $(x+y)^n$ by hand for larger n becomes a tedious task. The Edexcel Formula Booklet provides the following formula for binomial expansion:
$(a+b)^n=a^n+\left(\begin{array}{c}n\\1\end{array}\right)a^{n-1}b+\left(\begin{array}{c}n\\2\end{array}\right)a^{n-2}b^2+…+\left(\begin{array}{c}n\\r\end{array}\right)a^{n-r}b^r+…+b^n,\hspace{30pt}n\in {\mathbb N}$
where
$\left(\begin{array}{c}n\\r\end{array}\right)=\frac{n!}{(n-r)!r!}$
(see Pascal’s triangle notes for more on this) 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 $b$ – see Example 4. Notice that this question asks you for descending powers of $x$ so you may need to swap the variables accordingly. Note that when there are also coefficients inside the brackets, the coefficients in the expansion change dramatically from those given in Pascal’s triangle.
Relationship to Binomial 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. See Statistics Example to see this in practice.
Questions by Topic
Examples of Binomial Expansion
Expand $(x+y)^3$.
Using Example 1, expand $(x+y)^4$.
Find the first three terms, in descending powers of $x$, of the binomial expansion of $(2x+4)^5$.
(Statistics)
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.
- Given that ${{8}\choose{2}}=\frac{8!}{2!q!}$, write down the value of $q$.
- Given that the coefficient of $x^2$ in the expansion of $\left(p-\frac{x}{8}\right)^8$ is 28, find the value of $p$.
- Using the first three terms of a binomial expansion, estimate the value of $1.995^8$.
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