How many 4-digit even numbers greater than 5000 can be

Solution

ATQ, Digits available: 0, 1, 3, 5, 6, 8. 4-digit even number > 5000. Thousands digit options (first digit, non-zero, >5000): {5, 6, 8}. Units digit must be even: {0, 6, 8}. No repetition. Case 1: Thousands digit = 5 Remaining digits: {0,1,3,6,8}. Units digit can be 0, 6, or 8. If units = 0 → middle 2 digits from {1,3,6,8}: 4P2 = 12 If units = 6 → middle 2 from {0,1,3,8}: 4P2 = 12 If units = 8 → middle 2 from {0,1,3,6}: 4P2 = 12 Total for this case = 12 + 12 + 12 = 36. Case 2: Thousands digit = 6 Remaining digits: {0,1,3,5,8}. Units digit can be 0 or 8 (6 already used). If units = 0 → middle 2 from {1,3,5,8}: 4P2 = 12 If units = 8 → middle 2 from {0,1,3,5}: 4P2 = 12 Total = 24. Case 3: Thousands digit = 8 Remaining digits: {0,1,3,5,6}. Units digit can be 0 or 6. If units = 0 → middle 2 from {1,3,5,6}: 4P2 = 12 If units = 6 → middle 2 from {0,1,3,5}: 4P2 = 12 Total = 24. Grand total = 36 + 24 + 24 = 84.