What is the value of $1 + {{}^nP_2}/2 +{{}^nP_3}/3 + ~........... ~ {}^nP_n/n $
${}^nP_r = \frac {n!} {(n-r)!}$
Attempt: $1 + {{}^nP_2}/2 +{{}^nP_3}/3 + ~........... ~ {}^nP_n/n $
$= 1 + \frac {n! }{2.(n-2)!} + \frac {n! }{3.(n-3)!} + ........ + \frac {n! }{n.(n-n)!}$
$ = 1 + {}^nC_2 + {}^nC_3 . 2! +~~ ............~~+ {}^nC_n. (n-1)! $ .........(1)
Now, $(1+x)^n = {}^nC_0 + {}^nC_1 x + {}^nC_2x^2 + ............ + {}^nC_nx^n$
Diferentiating the above doesnt seem to work because in (1), every term has coffecient of the form ${}^nC_r r!$
Help shall be appreciated. Thanks