Expand ${\left(x-5\right)}^5$ using the binomial theorem.

The term in $x^4$ in $\left(2x-1\right){\left(x-5\right)}^5$ is $k{x}^4$, then find the value of $k$.

Show that the expansions of $(3-2x{)}^4$ and $(-2x+3{)}^4$ are the same.

Will $(b-ax{)}^n$ and $(-ax+b{)}^n$ always have the same expansion? Explain your answer.

Find the binomial power, in the form ${\left(a+b\right)}^n$, with expansion $27x^3-108x^{2}y+144x{y}^2-64y^3$.

Show that ${}^{n}C_0+{}^{n}C_1+{}^{n}C_2+\dots ..+{}^{n}C_n=2^n$.