Question

Expand the following binomial:
${(1-\frac{1}{x})}^{10}$

Answer 1

Using binomial expansion, we can write the expansion ${(1-\frac{1}{x})}^{10}$ as

$1-{}^{10}{C}_1\left(\frac{1}{x}\right)+{}^{10}{C}_2\left(\frac{1}{x^2}\right)-{}^{10}{C}_3\left(\frac{1}{x^3}\right)+{}^{10}{C}_4\left(\frac{1}{x^4}\right)-{}^{10}{C}_5\left(\frac{1}{x^5}\right)+{}^{10}{C}_6\left(\frac{1}{x^6}\right)-{}^{10}{C}_7\left(\frac{1}{x^7}\right)+{}^{10}{C}_8\left(\frac{1}{x^8}\right)$

$-{}^{10}{C}_9\left(\frac{1}{x^9}\right)+{}^{10}{C}_{10}\left(\frac{1}{x^{10}}\right)$

Which solves to
$1-\frac{10}{x}+\frac{45}{x^2}-\frac{120}{x^3}+\frac{210}{x^4}-\frac{252}{x^5}+\frac{210}{x^6}-\frac{120}{x^7}+\frac{45}{x^8}-\frac{10}{x^9}+\frac{1}{x^{10}}$