I recently came across a complicated sum while working on my homework. Usually I have no issues evaluating sums, but this one has stumped me. WolframAlpha managed to find a closed form solution, but I can't seem to work out how I should go about deriving it. Here's the sum and closed form from WolframAlpha:
$$\sum_{x=1}^y\frac{\left(\frac{5}{6}\right)^{x-1}}{(x-1)!(y-x)!}=\frac{\left(\frac{11}{6}\right)^{y-1}}{(y-1)!}$$
I have tried writing out the sum term by term to look for patterns. I noticed the factorials in the denominator seem to pair up sometimes, but I haven't been able to leverage that to any use. The main way I know to handle factorials in sums is to find a power series representation that matches the sum, but this sum doesn't seem to match anything I can find.
I appreciate any hints or solutions. Ideally, a solution without any prior knowledge of the result.
These should probably be enough.
Hint 1: Use the binomial theorem.
Hint 2: $\dfrac{1}{(x - 1)!(y - x)!} = \dfrac{1}{(y - 1)!}\displaystyle{y - 1 \choose x - 1}$
Hint 3: $\dfrac{11}{6} = 1 + \dfrac{5}{6}$