How many (7)-digit odd numbers greater than (5000000) can be formed from digits (0,1,2,3,4,5,6,7,8) without repetition?

Correct answer: (35280)

Direct answer: Option C, 35,280. The number must have seven digits, be greater than 5,000,000, and be odd. Its first digit can be 5, 6, 7, or 8. Count by cases because the first digit also affects the available odd final digits. Case 1: first digit 5 or 7, two choices. The last digit must be odd; after choosing the first digit, three odd digits remain. The five middle positions can then be filled in 7P5=7×6×5×4×3=2,520 ways. This gives 2×3×2,520=15,120. Case 2: first digit 6 or 8, two choices. All four odd digits remain available for the last place, and the middle positions again have 2,520 arrangements. This gives 2×4×2,520=20,160. Total =15,120+20,160=35,280. Option A, 30,240, misses some cases. Option B, 33,600, uses an incorrect odd-digit count. Option C is correct. Option D, 40,320, overcounts. Memory cue: when the first and last conditions interact, split into cases.