If `((n-1)!)/((n-4)!)`:`(n!)/((n-4)!)=1:9` find the value of `n`

(i) If (n!)/(2.(n-2)!): (n!)/(4!.(n-4)!) = 2:1 , find the valye of n. (ii) If ((2n)!)/(3!(2n-3)!): (n!)/(2!(n-2)!) = 44:3 , then find the value of n.

If "^(2n+1)P_(n-1):^(2n-1)P_n=3:5, then find the value of n .

If .^(n)P_(4): .^(n)P_(5) = 1:2 , find the value of n.

If (n !)/(2!(n-2)!) and (n !)/(4!(n-4)!) are in the ratio 2:1 , find the value of ndot

If (n!)/( 2!( n-2) !) and (eta !)/( 4!(n-4)!) are in the ratio 2: 1 , find the value of n.

If the G.M. of a and b is (a^(n)+b^(n))/(a^(n-1)+b^(n-1)) then find the value of n.

If .^(n)C_(r-1): .^(n)C_(r): .^(n)C_(r+1)=3:4:5 , find the values of n and r.

If P(n ,4)=20xxP(n ,2), then find the value of n

If P(n-1,\ 3): P(n ,4)=1:9 find n

Let S_(n)=""^(n)C_(0)""^(n)C_(1)+""^(n)C_(1)""^(n)C_(2)+…..+""^(n)C_(n-1)""^(n)C_(n). "If" (S_(n+1))/(S_(n))=(15)/(4) , find the sum of all possible values of n (n in N)