Binomial Expansion

Pascal’s triangle is made by putting 1s down the sides, then every number comes from adding together the two above it.

The numbers on the nth row are the coefficients of (1+x)^{n}

(1+x)^{0}=1

(1+x)^{1}=1+x

(1+x)^{2}=1+2x+x^{2}

(1+x)^{3}=1+3x+3x^{2}+x^{3}

(1+x)^{4}=1+4x+6x^{2}+4x^{3}+x^{4}

and so on.

The binomial coefficient \begin{pmatrix}n\\m\end{pmatrix}=\dfrac{n!}{m!(n-m)!} where m and n are whole numbers and n>m. We can also write them as \text{}^{n}C_{m}.

The binomial coefficients are the same as the numbers in Pascal’s triangle.

This means we can write our expansion as:

(1+x)^{n}=\begin{pmatrix}n\\0\end{pmatrix}+\begin{pmatrix}n\\1\end{pmatrix}x+\begin{pmatrix}n\\2\end{pmatrix}x^{2}+...+\begin{pmatrix}n\\n-1\end{pmatrix}x^{n-1}+\begin{pmatrix}n\\n\end{pmatrix}x^{n}