jmc

We start with the numbers that start with 1. There are 4 ways to pick the next digit, then 3 ways to pick the third digit, 2 ways to pick the fourth, and 1 to pick the last. Therefore, there are $4\cdot 3\cdot 2\cdot 1=24$ integers with 1 as the first digit. Similarly, another 24 have 2 as the first digit. That's 48 numbers so far, so we want the second smallest number that starts with 3. The smallest is 31245, and the next smallest is $\boxed{31254}$.