This quantity comes up sometimes:
$$\frac{\lambda ^ k}{k!}$$
For example, in $e^\lambda = \sum_{k=0}^\infty \frac{\lambda^k}{k!}$. I think in this case it comes up because of Taylor's Theorem.
Is there an intuitive way to understand what $\frac{\lambda ^ k}{k!}$ represents?
For example, does it represent some simple amount of something in an illustrative combinatoric scenario, which would make its meaning obvious?
I'm looking for answers that illustrate what this particular ratio signifies, rather than the meaning of more complex constructions containing it (without explanation of the ratio). If the illustration sets the ratio in a context, I'm looking for what role the ratio itself plays in that context.
Since you posted the question under "probability" tag here is a probabilistic motivation for this ratio. If $\lambda$ is the expected number of occurrences of some event in a given period of time, then the probability that exactly $k$ events occur is proportional to $\lambda^k/k!$ (assuming independence), i.e. if $X$ is the number of events per unit of time, then $$ \mathsf{P}(X=k)=e^{-\lambda}\frac{\lambda^k}{k!}. $$
The RHS can be viewed as the limit of the probability that exactly $k$ events occur in $n$ tries, where the probability of the occurrence of each event is $\lambda/n$ (each event is independent of others). Namely, letting $X_n$ be the number of events in $n$ tries, $$ \mathsf{P}(X_n=k)=\frac{n!}{k!(n-k)!}\left(\frac{\lambda}{n}\right)^k\left(1-\frac{\lambda}{n}\right)^{n-k}\to e^{-\lambda}\frac{\lambda^k}{k!} \quad\text{as}\quad n\to\infty. $$