# Combinatorial/probabilistic meaning/analogy for $x^n / n!$

The expression $$x^n / n!$$ appears in the infinite sum defining $$e^x$$ 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 consecutive 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 ($$e^x/\cos(x)/\sin(x)$$).

• There are extremely many. One of the best known is a derangement.
– whuber
Jul 5, 2023 at 14:31
• @whuber I do not understand what $x^n/n!$ has to do with derangements, can you clarify? Jul 5, 2023 at 15:54

If you divide every term by $$e^{x}$$ so the sum is $$1$$

then you get the Poisson distribution probability $$e^{-x} \dfrac{x^n}{n!}$$

of seeing $$n$$ items when the expected number of items is $$x$$.

• this @Henry still has the term e^−x . could you give some intuitive example of a set of calculations that converge to e^x? Jul 8, 2023 at 21:48

If $$X_1,X_2,\ldots,X_n$$ are i.i.d $$\text{Uniform}(0,1)$$ random variables, then (check this for details)

$$P\left(\sum_{i=1}^n X_i \le x\right)=\frac{x^n}{n!}\quad,\, x\in [0,1]$$

In other words, $$x^n/n!$$ is the volume of the region $$\left\{(x_1,\ldots,x_n)\in \mathbb R^n: x_1,\ldots,x_n\ge 0, \sum_{i=1}^n x_i \le x\right\}$$ whenever $$x\in [0,1]$$. When $$x=1$$, this is related to the volume of a simplex.

We can also define a stopping rule associated with the $$X_i$$'s as

$$N=\min\left\{n: \sum_{i=1}^n X_i>x\right\} \quad,\, x\in [0,1]$$

Then, using the result above,

$$E(N)=\sum_{n=0}^\infty P(N>n)=\sum_{n=0}^\infty P\left(\sum_{i=1}^n X_i \le x\right)=e^x$$

This is in fact a popular exercise discussed here previously.

• What I like about this is the result is simple and yet the interpretation cannot be more manifesting. +1. Jul 6, 2023 at 6:22
• @StubbornAtom what is the meaning of E(N)? Jul 8, 2023 at 21:46
• @Gilad It is the expected number of $X_i$s when their cumulative sum first exceeds the given $x$ Apr 2 at 10:23