Question Number 115170 by bobhans last updated on 24/Sep/20

(1)Given ((P _(n−1)^(2n+1) )/(P _n^(2n−1) )) = (3/5) , find n = ?  (2) in how many ways can 6 persons  stand in a queue?  (3) how many different 4 letter words  can be formed by using letters of   EDUCATION using each letter at   most once ?

Answered by bemath last updated on 24/Sep/20

(1) ((((2n+1)!)/((n+2)!))/(((2n−1)!)/((n−1)!))) = (3/5) → (((2n+1)!(n−1)!)/((2n−1)!(n+2)!)) = (3/5)      (((2n+1).2n.(n−1)!)/((n+2)(n+1)n(n−1)!)) = (3/5)       ((4n+2)/((n+2)(n+1))) = (3/5)       20n+10 = 3(n^2 +3n+2)       3n^2 −11n−4 = 0  (3n+1)(n−4) = 0 → n=4

Answered by bemath last updated on 24/Sep/20

(2) ^6 C_6  ×6! = 720

Answered by bemath last updated on 24/Sep/20

(3) ^9 C_4 ×4! = ((9!)/(4!.5!)) ×4! = ((9!)/(5!))