If [ n+1 r+1 ]: [ ^n C_r ]: [ n-1 r-1 ]=11: 6: 3 then n r=

Step-by-step solution

Given the ratio ^ n+1 C_ r+1 : ^ n C_ r : ^ n-1 C_ r-1 = 11 : 6 : 3 . First, consider the ratio ^ n+1 C_ r+1 ^ n C_ r = 11 6 . Using the formula ^ n C_ r = n! r!(n-r)! , we have (n+1)! (r+1)!(n-r)! r!(n-r)! n! = n+1 r+1 = 11 6 . This implies 6n + 6 = 11r + 11 , or 6n - 11r = 5 . (Equation 1) Next, consider the ratio ^ n C_ r ^ n-1 C_ r-1 = 6 3 = 2 . Using the formula, n! r!(n-r)! (r-1)!(n-r)! (n-1)! = n r = 2 . This implies n = 2r . (Equation 2) Substituting n = 2r into Equation 1: 6(2r) - 11r = 5 12r - 11r = 5 r =