Question

If 3(nP2 + 24)=2nP2, find the positive value of n​

Answer 1

`n = 8`

Explanation:

Given

`3(^nP_2 + 24) = ^(2n)P_2`

Required

Find n

To do this, we simply apply permutations formula

`nP_r = (n!)/((n -r)!)`

So, we have:

`3 * [(n!)/((n -2)!) + 24] = (2n!)/((2n -2)!)`

Expand

`3 * [(n * (n - 1) * (n - 2)!)/((n -2)!) + 24] = (2n * (2n - 1) * (2n - 2))/(2n - 2)`

`3 * [n * (n - 1) + 24] = 2n * (2n - 1)`

`3 * [n^2 - n + 24] = 4n^2 - 2n`

Open bracket

`3n^2 - 3n + 72 = 4n^2 - 2n`

Collect like terms

`3n^2 - 4n^2- 3n+2n + 72 = 0`

`-n^2- n + 72 = 0`

Expand

`-n^2 -9n + 8n + 72 = 0`

Factorize

`-n(n +9) - 8(n + 9) = 0`

Factor out n + 9

`(-n -8)(n + 9) = 0`

Split

`(-n -8)= 0 \ or\ (n + 9) = 0`

Solve for n

`n =8\ or\ n = -9`

The positive value is
`n = 8`