Question

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

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

Answer 1

`n=6`

Explanation:

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

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

`(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`.