what is exponential generating function with n Choose k as coefficient

Question

If we fix a positive integer $k$, what is the EGF of $\sum_{n=0} \binom{n}{k} \frac{x^n}{n!}$ ?

I know EGF of $\sum_{n=0}\frac{x^n}{n!}$ is $e^x$, but the addition of $\binom{n}{k}$ confuses me

Answer 1

Well, $$\sum_{n=0}^\infty\binom nk\frac{x^n}{n!}=\sum_{n=k}^\infty\binom nk\frac{x^n}{n!}=\frac1{k!}\sum_{n=k}^\infty\frac{x^n}{(n-k)!}=\frac{x^ke^x}{k!}.$$