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The expression $x^n / n!$ appears in a the infinite sum definig $e^x$ and similar terms in the sums defining $\cos(x)$, $\sin(x)$, etc. I would like to know if there is some combinatorial/probabilistic meaning or analogy to the term $x^n / n!$ and appropriately an example of a scenario of selection or decision making that could be represented by a consequtive sum of these terms similar to the infinite sums defining $e^x$ or $\cos(x)/\sin(x)$.

In other words: some probabalistic/Combinatorical scenario whose calculation would converge to one of these functions ($e^x/\cos(x)/\sin(x)$). Thanks alot

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  • $\begingroup$ I am not qualified to really talk about this, but this might be of interest. There is a category theoretic/algebraic idea called groupoid cardinality. Taken from Qiaochu Yuan's blog, "Let $s$ be a finite set and consider the groupoid of $s$-colored finite sets ... the groupoid cardinality is $\sum_{n\ge0 }\frac{|s|^n}{n!}$." $\endgroup$ Commented Jul 5, 2023 at 13:55
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    $\begingroup$ Cross-posted at stats.stackexchange.com/q/620593/119261. $\endgroup$ Commented Jul 5, 2023 at 15:36

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