Given $N$ blocks (numbered $1$ to $N$) and an array with $L$ slots, $N > L$, I need to find out how many possibilities are there to arrange the blocks in the array without repeating any block. The slots in the array can also be empty - in this case they will be labeled $0$.
For example for $N=7$ and $L=5$ some valid combinations are $[4,7,2,5,1]$, $[1,0,0,7,3]$ or $[1,7,3,0,0]$ but not $[1, 4, 4, 5, 6]$.
I came up with the following result $$\sum^L_{k=0} P(N,L-k)C(L,k)$$ Where P(,) denotes permutation and C(,) combinations.
Is this correct? Does this simplify to a more concise formula?
Edit: If a concise formula is not available is it possible to find a lower bound to the result of the exponential form? i.e. $L^x$ or $N^x$