If 5 out of 10 students are freshmen in the population of students, then the probability p of a student being a freshman=½, which is the same probability of the student not being a freshman.
(½+½)6=(½)6(1+6+15+20+15+6+1) is a binomial expansion which determines the individual probabilities of 0, 1, 2, 3, 4, 5, 6 students in a random sample of 6 students. For example, the probability of 2 out of 6 being freshmen is the third term in the series, which is 15/64 (because 26=64).
So if s represents the number of students, where 0≤s≤6, then:
p(s)=6Cs/64 or 6!/(s!(6-s!))/64=(45/4)/(s!(6-s!)) is the probability distribution formula.