For any finite non empty set X of integers, let max(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 max(A) × |B| = 12; and |A| × max(B) = 11 and N can be written as 100a + $b$ where a, $b$ are positive integers less than 100, find $a$ + b

Step 1: A = {a1, a2, a3……ap}

Step 2: B = {b1, b2, b3….bq}

Step 3: Case-A : $p$ = 11, bq = 1

Step 4: A = {a1, a2, a3….a11}, B = {1}

Step 5: => a11 = 12, $q$ = 1

Step 6: => 11C

Step 7: 10 = total ways

Step 8: Case-B : $p$ = 1, bq = 11

Step 9: (1) A = {12}, B = {11} → 1 way

Step 10: (2) A = {6}, B = {b1, 11} → 10C1 ways

Step 11: (3) A = {4}, B = {b, b2, 11} → 10C2 ways

Step 12: (4) A = {3}, B = {b1, b2, b3, 11} → 10C3 ways

Step 13: (5) A = {2}, B = {b1, b2, b3, b4, b5,11} → 10C5 ways

Step 14: (6) A = {1}, B = {b1, b2, …..b11, 11} → 0 ways

Step 15: => Total ways = 11 + 1 + 10 + 45 + 120 + 252

Step 16: = 100 × 4 + 39

Step 17: a + $b$ = 43