Question

Prove that `""^(n+1)P_r=((n+1))/((n-r+1))xx""^nP_r`?

Answer 1

Please see below.

Explanation:

The formula for the number of possible permutations of `r` objects from a set of `n` is written as `""^nP_r` and is

`""^nP_r=(n!)/((n-r)!)`,

where `k!` is defined as `k! =kxx(k-1)xx(k-2)...3xx2xx1`

Hence `""^(n+1)P_r=((n+1)!)/((n+1-r)!)=((n+1)!)/((n-r+1)!)`

= `((n+1)xxn!)/((n-r+1)xx(n-r)!)`

= `((n+1))/((n-r+1))xx(n!)/((n-r)!)`

= `((n+1))/((n-r+1))xx""^nP_r`