Problem 8 · Block-swap pairing with a leading-zero check · 5 points
Made by AI and not reviewed by a real person.
Using 0, 1, 2, 3, and 4 exactly once, make five-digit numbers whose last two digits also form a two-digit number. How many have a larger two-digit number at the start than at the end?
- 12
- 48
- 72
- 60
- 36
Show a hint
- What happens if you swap the two-digit block at the start with the one at the end? Check whether the new number is still allowed.
- Where can 0 go without starting either two-digit block? Then count the ways to arrange the other digits.
Show the answer and solution
Answer: E — 36
The 0 cannot be first or fourth, so it has 3 possible positions. The other digits have 4 × 3 × 2 × 1 = 24 orders, making 72 allowed numbers. Swapping the two end blocks pairs these numbers: exactly one in each pair has the larger block at the start, since the blocks cannot be equal. So the answer is 72 ÷ 2 = 36. The tempting 48 comes from halving all 96 five-digit arrangements, including ones where 0 starts the last block and it is not a two-digit number.