The value of $\sum ^n_{r=1}\frac{{{{}^nP}}_r}{r!}$ is:

Question: Find the value of:

$$ \sum_{r=1}^{n} \frac{nP_r}{r!} $$

Solution:

We know: \( nP_r = \frac{n!}{(n - r)!} \Rightarrow \frac{nP_r}{r!} = \frac{n!}{(n - r)! \cdot r!} = \binom{n}{r} \)

Therefore,

$$ \sum_{r=1}^{n} \frac{nP_r}{r!} = \sum_{r=1}^{n} \binom{n}{r} = 2^n - 1 $$

Final Answer: $$ \boxed{2^n - 1} $$