Question

Binomial theorem problem in comments (need correct markup). Solved a bit of it, need help with the rest.

Answer 1

If 2 <= k <= n - 2, show that
\[\left(\begin{matrix}n \\ k\end{matrix}\right) = \left(\begin{matrix}n - 2 \\ k -2\end{matrix}\right) + 2\cdot\left(\begin{matrix}n-2 \\ k-1\end{matrix}\right) + \left(\begin{matrix}n-2 \\ k\end{matrix}\right)\]
for n >= 4

With Pascal's theorem I got so far: \[\left(\begin{matrix}n \\ k\end{matrix}\right) = \left(\begin{matrix}n - 2 \\ k -2\end{matrix}\right) +\left(\begin{matrix}n-2 \\ k-1\end{matrix}\right) + \left(\begin{matrix}n-2 \\ k-1\end{matrix}\right) + \left(\begin{matrix}n-2 \\ k\end{matrix}\right) = \left(\begin{matrix}n - 1 \\ k-1\end{matrix}\right) + \left(\begin{matrix}n - 1 \\ k+ 1\end{matrix}\right)\]

What should I do next? For limitations I got that for the first binomial it should be 2 <= k <= n + 1 and for the second 1<= k + 1 <= n - 2.

Answer 2

@welshfella maybe you could help me out here? :)