# 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.

### 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:

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

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.

## Examples of Binomial Expansion

Expand $(x+y)^3$.

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

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

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.

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 $1.995^8$.