Answer:
[tex]^nP_r=n![/tex] Proved.
Step-by-step explanation:
Consider the provided information.
We need to explain why [tex]^nP_n=n![/tex] for all positive integers n.
The permutation formula is: [tex]^nP_r=\frac{n!}{(n-r)!}[/tex]
Now substitute r=n in above formula as we want the value of [tex]^nP_n[/tex]
[tex]^nP_n=\frac{n!}{(n-n)!}[/tex]
[tex]^nP_r=\frac{n!}{(0)!}[/tex]
The value of 0! is 1.
Therefore,
[tex]^nP_r=\frac{n!}{1}[/tex]
[tex]^nP_r=n![/tex]
Hence, proved.
