For any finite non empty set X of integers, let (X) denote the largest element of X and |X| denote the number of elements in X . If N is the number of ordered pairs ( A, B ) of finite non-empty sets of positive integers, such that aligned & (A) |B|=12 ; and |A| (B)=11 aligned and N can be written as 100 a+b where a, b are positive integers less than 100 , find a+b

Step-by-step solution

aligned & A= a₁, a₂, a₃ a_p & B= b₁, b₂, b₃ . b_q & aligned & a_p q=12 & p b_q=11 aligned & Case-A : p=11, b_q=1 & A= a₁, a₂, a₃ . a₁₁ , B= 1 & a₁₁=12, q=1 & 11 10 = total ways & Case-B : p=1, b_q=11 & (1) A= 12 , B= 11 1 way & (2) A= 6 , B= b₁, 11 ¹⁰ C₁ ways & (3) A= 4 , B= b₁, b₂, 11 ¹⁰ C₂ ways & (4) A= 3 , B= b₁, b₂, b₃, 11 ¹⁰ C₃ ways & (5) A= 2 , B= b₁, b₂, b₃, b₄, b₅, 11 ¹⁰ C₅ ways & (6) A= 1 , B= b₁, b₂, . b₁₁, 11 0 ways & Total ways =11+1+10+45+120+252 & =439 & =100 4+39 aligned