I'm looking for a simple identity for the formula:

$$ \sum_{n = 0}^{p} \binom{p}{n} \cdot n! \cdot x^n $$

In words, I have $p$ "players" who can choose to play or not (every player is represented by a unique id). Those who chose to play are lined up in all possible orders. Then every playing player picks an element out of $x$ possibilities (with repetition allowed), in addition to his id.

How many sequences can we get? Is there a simple solution for this series? If not, what is the closest upper limit you can think of?

I haven't touched combinatorics for a long time, so there could be a simple identity that I'm missing...