Answer
The statement is true.
Work Step by Step
Permutation problems are arrangement problems which can be solved by observing a
successive list of questions:
1. In how many ways can we choose the 1st object? Answer: $n$.
2. In how many ways can we choose the 2nd object? Answer: $n-1$.
...
r. In how many ways can we choose the rth object? Answer: $n-r+1$.
The total number of ways is, using FCP,
$n(n-1)n-2)\cdot...\cdot(n-r+1)$,
The formula for ${}_{n}P_{r}$,
(the number of ordered sequences taking r objects from n,
each not being used more than once.),
leads to the same expression :
${}_{n}P_{r}=\displaystyle \frac{n!}{(n-r)!}=$
$=\displaystyle \frac{n(n-1)\cdot...\cdot ...(n-r+1)(n-r)!}{(n-r)!}$
... after reducing $(n-r)!$, ...
$= n(n-1)(n-2)...(n-r+1)$
The statement is true.
