Let N be the number of ways of distributing 52 identical balls into 4 distinguishable boxes such that no box is empty and the difference between the number of balls in any two of the boxes is not a multiple of 6 . If N=100 a+b , where a, b are positive integers less than 100 , find a+b .

Step-by-step solution

Let i^ h box has 6 _ i + _ i balls where _i, _i w and _i 5 . Also, all _ i 's are distinct. 6 ( ₁+ ₂+ ₃+ ₄ )+ ( ₁+ ₂+ ₃+ ₄ )=52 and ₁+ ₂+ ₃+ ₄ 6,7,8,9,10,11,12,13,14 Hence, only one possibility is there array ll & ₁+ ₂+ ₃+ ₄=7 ...(i) and & ₁+ ₂+ ₃+ ₄=10...(ii) array Equation (ii) has only three solution sets, which are (5,4,1,0),(5,3,2,0) and (4,3,2,1) . Equation (i) has total ¹⁰ C₃ solutions and ^4 C₁ ^9 C₂ solutions when any _i is zero. So, N=3 ( ¹⁰ C₃ (4!) )-2 ( ^4 C₁ ^9 C₂(3!) )=6912 a=69 and b=12 .