# Find n if ""^nP_3=20n?

## The answer is 6 but I'm unsure how to get to it.

Oct 11, 2017

$n = 6$

#### Explanation:

""^nP_r is defined as (n!)/(r!).

Thus, ""^nP_3=(n!)/(3!). We know that this is equal to $20 n$.

(n!)/(3!)=20n

Multiply both sides by (3!)/n:
(n!)/n=20*3!

We know that 3! =3*2*1=6, and n!=n*(n-1)!.

(n*(n-1)!)/n=20*6
(n-1)! =120

Since 5! =5*4*3*2*1=120, $n - 1 = 5$.

The final answer is $n = 5 + 1 = 6$.