Find n and r if `.^(n)C_(r):^(n)C_(r+1):^(n)C_(r+2)=1:2:3`.

Find n and r if ^(n)C_(r-1) : ^(n)C_r : ^(n)C_(r+1) = 14 : 8 : 3

Find n and r if .^(n)P_(r)=.^(n)P_(r+1)and^(n)C_(r)=^(n)C_(r-1) .

Find n if, ^(2n)C_(r-1) = ^(2n)C_(r+1)

If ""^(n)C_(r ): ""^(n)C_(r+1)=1:2 and ""^(n)C_(r+1): ""^(n)C_(r+2)=2:3 , find n and r.

nC_(r):^(n)C_(r+1)=1:2 and ^(n)C_(r+1):^(n)C_(r+2)=2:3, findn and r

Prove that: (i) r.^(n)C_(r) =(n-r+1).^(n)C_(r-1) (ii) n.^(n-1)C_(r-1) = (n-r+1) .^(n)C_(r-1) (iii) .^(n)C_(r)+ 2.^(n)C_(r-1) +^(n)C_(r-2) =^(n+2)C_(r) (iv) .^(4n)C_(2n): .^(2n)C_(n) = (1.3.5...(4n-1))/({1.3.5..(2n-1)}^(2))

If .^(n+1)C_(r+1):^(n)C_(r):^(n-1)C_(r-1)=11:6:3 , find the values of n and r.