If the number of permutations of `n` different things taken `r` at a time be denoted by `nP_r` show that `(nP_1)/(1!)+(nP_2)/(2!)+(nP_3)/(3!)+......+(nP_n)/(n!)=2^n-1`

Let nP_(r) denote the number of permutations of n differenttaken r at a time.Then,prove that 1+1*1P_(1)+2*2P_(2)+3*3P_(3)+......+n*nP_(n)=n+1P_(n+)

Statements-I:: The no. of permutations of n different things taken any no. of things at a time is nP_n, (repetition not allowed) Statements-II : The no. of permutations of n different things taken any no. of things at a time is nP_1+nP_2 +nP_3+ +nP_n (repetition not allowed)

Show that ^(^^)nP_(r)=(n-r+1)^(n)P_(r-1)

Simplify (nP_4)/((n-1)P_3

Prove that ^(n-1) P_r+r .^(n-1) P_(r-1) = .^nP_r

Prove that P(n;r)=nP_(r)=n(!)/(n-r)!

11. Prove that nP_(r)=n(n-1)P_(r-1)

Prove that .^(n-1) P_r+r .^(n-1) P_(r-1) = .^nP_r

nP_ (n) = np_ (n-1) and nP_ (r) = rP_ (n) * nP_ (nr)

Prove that: ""^(n+1)P_(r+1)=(n+1)* ""^nP_r .