Incorrect. My solution counted permutations rather than combinations. The number of combinations is small enough that they can quickly be listed and counted.
Incorrect. At the time, I could not figure out how to solve the problem. In hindsight, I could have solved it by using one of my favorite techniques: a table diagram. In the image below, I constructed a 10 x 10 table to see which of the first 10 lockers are closed after the first ten students exited the school. The result is that only the lockers numbered as perfect squares were closed. Assuming that this pattern continues to the 900th locker, it is a safe guess that all lockers numbered as perfect squares will be the only ones that are closed. It is also interesting to observe that the total number of lockers, 900, is a perfect square. The square root of 900 is 30, which means there are a total of 30 perfect squares from 1 to 30.