# combinatorical\probabalistic interpretation of $(x^n)/n!$

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

• 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 $\sum_{n\ge0 }\frac{|s|^n}{n!}$." Commented Jul 5, 2023 at 13:55
• Cross-posted at stats.stackexchange.com/q/620593/119261. Commented Jul 5, 2023 at 15:36