Binomial Theorem

What is the Binomial Theorem Formula?

The binomial theorem allows us to expand an expression of the form ${(x+y)}^n$ into a polynomial sum containing x and y terms. A general formula for a binomial expression is given by,

${(x+y)}^n=\left(\begin{array}{c}n\\ 0\end{array}\right)x^n{y}^0+\left(\begin{array}{c}n\\ 1\end{array}\right)x^{n-1}{y}^1+\left(\begin{array}{c}n\\ 2\end{array}\right)x^{n-2}{y}^2+...+\left(\begin{array}{c}n\\ n-1\end{array}\right)x^1{y}^{n-1}+\left(\begin{array}{c}n\\ n\end{array}\right)x^0{y}^n$.

which can be simplified to the following.

In the above expression, $\sum _{k=0}^n$denotes the sum of all the terms starting at k = 0 until k = n.

Note that x and y can be interchanged here so the binomial theorem can also be written a

${(x+y)}^n=\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)x^k{y}^{n-k}$

and this would give the same results as above.

The notation $\left(\begin{array}{c}n\\ k\end{array}\right)$ can be referred to as "n choose k" and gives a number called binomial coefficient, which is the number of different combinations of ordering k objects out of a total of n objects.

For example for 5 choose 3, we would have,

.

What is the proof for the binomial theorem?

The binomial theorem can be proved in a few different ways, but we will focus on the combinatorial proof.

Any expression in the form of ${(x+y)}^n$ can be written as

${(x+y)}^n=(x+y)\times (x+y)\times ...\times (x+y)$,

with a total of n products.

After expanding and removing the parenthesis, each term has the form of $x^{n-k}{y}^k$, for some arbitrary k between 0 and n.

The coefficient of this term has to be the number of ways to choose k values of y out of n values of $(x+y)$. Therefore, the coefficient of $x^{n-k}{y}^k$ is $\left(\begin{array}{c}n\\ k\end{array}\right)$ and

${(x+y)}^n=\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)x^{n-k}{y}^k$.

How to do a binomial expansion?

To understand how to expand a binomial expansion, we will look at an example.

Let's say we want to expand . In this case, n = 4 and k will vary between 0 and 4. Using the formula for the binomial theorem, we can write:

.

We can now use the equation for the binomial coefficient to find all the constant terms in this expression. For the first term, 4 choose 0 (4C₀), we have:

.

Repeating this for all 5 coefficients, we end up with binomial coefficients of ⁴C₀ = 1, ⁴C₁ = 4, ⁴C₂ = 6, ⁴C₃ = 4 and ⁴C₄ = 1 in order.

Therefore, our expression for the binomial expansion simplifies to:

.

Binomial theorem examples

The binomial theorem can also be used to find a specific term for a binomial expansion. For this, you do not have to carry out the whole expansion but will only be required to find one term. Let's look at an example to see how this can be done.