Show-that-for-n-N-r-0-n-P-r-n-n-e-where-x-denotes-the-greatest-integer-x-and-P-r-n-n-n-r-

Question Number 55099 by Joel578 last updated on 17/Feb/19

Show that for n ∈ N, Σ_(r=0) ^n P_r ^n = ⌊n! e⌋ where ⌊x⌋ denotes the greatest integer ≤ x and P_r ^n = ((n!)/((n − r)!))

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{for}\:{n}\:\in\:\mathbb{N}, \\ $$$$\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\:{P}_{{r}} ^{{n}} \:=\:\lfloor{n}!\:{e}\rfloor \\ $$$$\mathrm{where}\:\lfloor{x}\rfloor\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer}\:\leqslant\:{x} \\ $$$$\mathrm{and}\:{P}_{{r}} ^{{n}} \:=\:\frac{{n}!}{\left({n}\:−\:{r}\right)!} \\ $$

Answered by tm888 last updated on 18/Feb/19