Ernest Ryland

2021-12-14

Binomial expansion of ${\left(1-x\right)}^{n}$
Weve

Deufemiak7

Well, as I understand it, we could write the binomial expansion as:
$\left(1-x{\right)}^{n}=\sum _{k=0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){1}^{n-k}\left(-x{\right)}^{k}$
$\left(\begin{array}{c}n\\ 0\end{array}\right){1}^{n}\left(-x{\right)}^{0}+\left(\begin{array}{c}n\\ 1\end{array}\right){1}^{n-1}\left(-x\right)+\left(\begin{array}{c}n\\ 2\end{array}\right){1}^{n-2}\left(-x{\right)}^{2}+\left(\begin{array}{c}n\\ 3\end{array}\right){1}^{n-3}\left(-x{\right)}^{3}...$
which simplifies to $1-nx+\frac{n\left(n-1\right)}{2!}\cdot {x}^{2}-\frac{n\left(n-1\right)\left(n-2\right)}{3!}\cdot {x}^{3}\dots$
Which is the answer everyone else has given.

maul124uk