Let S be the set of all 4 -digit numbers (in base 10 ) d₁ d₂ d₃ d₄ , where d₁ is the thousands digit, such that the sum of their digits is 15 . If it is given that d₁ 2 , d₂ 4 , and d₃ 5 , then the number of elements in S is equal to

Step-by-step solution

Let the 4 -digit number be d₁ d₂ d₃ d₄ . Given the sum of the digits is 15 , we have d₁ + d₂ + d₃ + d₄ = 15 . The constraints on the digits are: 2 d₁ 9 0 d₂ 4 5 d₃ 9 0 d₄ 9 Let y₁ = d₁ - 2 , y₂ = d₂ , y₃ = d₃ - 5 , and y₄ = d₄ . Substituting these into the sum equation gives: y₁ + y₂ + y₃ + y₄ = 15 - 2 - 5 = 8 The new constraints are: 0 y₁ 7 0 y₂ 4 0 y₃ 4 0 y₄ 9 The number of valid numbers is the coefficient of x^8 in the expansion of: (1 + x + + x^7)(1 + x + + x^4)^2(1 + x + + x^9) This can be written using genera