${\left(x\+y\right)}^5$

$\=\sum _{k\=0}^5\left(\genfrac{}{}{0ex}{}{5}{k}\right) x^{5-k} y^k$

$\=\left(\genfrac{}{}{0ex}{}{5}{0}\right)x^5{y}^0\+\left(\genfrac{}{}{0ex}{}{5}{1}\right)x^4{y}^1\+\left(\genfrac{}{}{0ex}{}{5}{2}\right)x^3{y}^2\+\left(\genfrac{}{}{0ex}{}{5}{3}\right)x^2{y}^3\+\left(\genfrac{}{}{0ex}{}{5}{4}\right)x^1{y}^4\+\left(\genfrac{}{}{0ex}{}{5}{5}\right)x^0{y}^5$

$\=\frac{5\!}{0\! 5\!}{x}^5\+\frac{5\!}{1\! 4\!}{x}^{4}y\+\frac{5\!}{2\! 3\!}{x}^3{y}^2\+\frac{5\!}{3\! 2\!}{x}^2{y}^3\+\frac{5\!}{4\! 1\!}{\mathrm{xy}}^4\+\frac{5\!}{5\! 0\!}{y}^5$

$\=x^5\+5x^{4}y\+10x^3{y}^2\+10x^2{y}^3\+5x{y}^4\+y^5$

${\left(x\+y\right)}^5\=\sum _{k\=0}^5\left(\genfrac{}{}{0ex}{}{5}{k}\right) x^{5-k} y^k\=x^5\+5x^{4}y\+10x^3{y}^2\+10x^2{y}^3\+5x{y}^4\+y^5$