In how many ways can one climb a staircase composed of 10 stairs when each step covers either one stair or two stairs?
This question is very similar to one I have seen with 6 stairs. The solution was to find how many possibilities are there to take one stair at a time, then 2, then 3, and so on. I need help because the number is larger.
$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
\begin{align} &\sum_{s_{1} = 1}^{2}\ldots\sum_{s_{n} = 1}^{2} \bracks{s_{1} + \cdots + s_{n} = 10} = \sum_{s_{1} = 1}^{2}\ldots\sum_{s_{n} = 1}^{2} \oint_{\verts{z} = 1}{1 \over z^{11 - s_{1} - \cdots - s_{n}}} \,{\dd z \over 2\pi\ic} \\[5mm] = &\ \oint_{\verts{z} = 1}{1 \over z^{11}}\pars{\sum_{s = 1}^{2}z^{s}}^{n} \,{\dd z \over 2\pi\ic} = \oint_{\verts{z} = 1}{\pars{1 + z}^{n} \over z^{11 - n}}\,{\dd z \over 2\pi\ic} = {n \choose 10 - n} \end{align} which is non zero whenever $\ds{\pars{~n \geq 10 - n \geq 0 \implies 5 \leq n \leq 10~}}$.
$$ \sum_{n = 5}^{10}{n \choose 10 - n} = \sum_{n = 0}^{5}{n + 5 \choose 5 - n} = \bbx{\ds{89}} $$