Out of the 69 subjects, 20 are subjects you want to study: call these "likes". The remaining 49 are subjects you do not want to study: call these "dislikes".
5 subjects are chosen at random out of the 69. Probability p that a subject is a like is 20/69, and probability 1-p=49/69 that it's a dislike. Probability that all 5 randomly chosen subjects are all likes is (20/69)5=0.002046 approx. or about 0.2% (1 in 500). The binomial expansion of (p+(1-p))5=1 covers all possible outcomes:
p5+5p4(1-p)+10p3(1-p)2+10p2(1-p)3+5p(1-p)4+(1-p)5=1 is:
all likes; 4 likes, 1 dislike; 3 likes, 2 dislikes; 2 likes, 3 dislikes; 1 like, 4 dislikes; all dislikes. The terms in the expansion represent the probabilities described in order. For example, the probability that 2 of the randomly chosen subjects are likes is about 30%.
The probability that at least one of the 5 random subjects is a like is 1-(1-p)5=1-0.1806=0.8194 or almost 82%.
I'm not sure if this is what your question is all about, but hopefully this proposed solution is helpful.