The number of ways to distribute 20 identical candies among 4 children C₁, C₂, C₃ , and C₄ such that C₁ receives at most k candies, C₂ receives at most 5 candies, C₃ receives at least 2 candies, and C₄ receives at least 1 candy is 660 . The value of k is

Step-by-step solution

Let the number of candies received by the children be C₁, C₂, C₃, C₄ . Given C₁ + C₂ + C₃ + C₄ = 20 . The constraints are: 0 C₁ k 0 C₂ 5 C₃ 2 C₄ 1 Let y₃ = C₃ - 2 0 and y₄ = C₄ - 1 0 . Substituting these into the equation gives: C₁ + C₂ + y₃ + y₄ = 20 - 2 - 1 = 17 The number of ways is the coefficient of x¹⁷ in the generating function: (1+x+ +x^k)(1+x+ +x^5)(1+x+ )^2 = ( 1-x^ k+1 1-x ) ( 1-x^6 1-x ) (1-x)⁻² = (1-x^ k+1 )(1-x^6)(1-x)⁻⁴ = (1 - x^6 - x^ k+1 + x^ k+7 )(1-x)⁻⁴ The coefficient of x¹⁷ is: ¹⁷⁺⁴⁻¹C₁₇ - , ¹¹