Question

If ⁿP_r = 1680 and ⁿC_r = 70, find n and r.

Answer 1

ⁿP_r = 1680, ⁿC_r = 70, n = ? p = ?

ⁿC_r  = `1/(r!)` ⁿP_r

`70/1680 = 1/(r_!)`

`1/24 = 1/(r_i)`

r! = 24

r! = 4 × 3 × 2 × 1

r! = 4!

r = 4

ⁿP_r  = 1680

`(n!)/(n - 4) = 1680`

`(n (n  - 1)(n - 2) (n - 3) (n - 4)!)/(n - 4!) = 1680`

n (n - 1) (n - 2) (n - 3) = 1680

n = 8, 8 (8 - 1) (8 - 2) (8 - 3) = 1680

n = 8, r = 4