Let M be the maximum value of the ratio ¹⁴C_r ¹⁶C_ r+1 over all possible valid integers r . If M can be expressed as p q where p and q are coprime positive integers, then the value of p+q is :

Step-by-step solution

The given ratio is ¹⁴C_r ¹⁶C_ r+1 . Expanding the combinations into factorials, we get: ¹⁴C_r ¹⁶C_ r+1 = 14! r!(14-r)! 16! (r+1)!(15-r)! ¹⁴C_r ¹⁶C_ r+1 = 14! 16! (r+1)! r! (15-r)! (14-r)! ¹⁴C_r ¹⁶C_ r+1 = (r+1)(15-r) 15 16 = (r+1)(15-r) 240 For the combinations to be defined, we must have 0 r 14 . To maximize the ratio, we need to maximize the numerator f(r) = (r+1)(15-r) . This is a downward-opening parabola with its vertex at r = -1 + 15 2 = 7 . Since r = 7 is an integer within the valid domain, the maximum value