Well, first of all, let me just say that determining the percentage of coffee drinkers in your university sounds like a really stimulating task!
To answer your question, if we assume that Y follows a binomial distribution (with probabilities p for success and 1-p for failure), then we only need to determine the value of one unknown: p. This is because the binomial distribution is completely specified by the number of trials (n) and the probability of success in each trial (p).
So, to summarize, you only need to determine the value of one unknown (p) to fully specify the distribution of Y. Just like a good cup of coffee, one unknown is all you need to get a satisfying answer!