Random sampling job applicants

Recently, one of my friends told me of her job application, which had been rejected before even the second stage of the recruitment process, which was an entirely automated cognitive aptitude test. I joked that maybe the company randomly sampled their applicants, due to high demand for employment. This made me wonder, however, whether such a procedure was in fact reasonable. After all, processing candidates is expensive and time-consuming, right?

Consider a job application process: there are a certain number of applicants n, of which only some proportion p are “suitable” for the position (in this case, we will consider all suitable candidates equally so). Fortunately, the company only has some t < n * p positions open, and thus does not need to find all the suitable candidates. Say they wanted to have a 90% chance of finding at least t suitable candidates—how much of the applicant population would they need to sample?

Of the answer, I’m not sure… perhaps someone can enlighten me. An interesting question though, no?