Question

Find k in the expansion (m-2n)^10 = m^10 - 20m^9n + km^8n^2 - .... + 1024n^10

Answer 1

note that\[(m-2n)^{10}=\sum_{l=0}^{10} \left(\begin{matrix}10 \\ l\end{matrix}\right) m^{10-l} (-2n)^l\]

Answer 2

Use binomial expansions.
\[\left(\begin{matrix}10 \\ 2\end{matrix}\right)(-2)^2 = k\]