Proof of Bernoulli trials

Question

I am studying Grinstead and Snell "Introduction to probability", but I am stuck at the proof of the Bernulli trials. I understood that (n,j) can be the number of j-size subsets (whatever the order) of n-size sets, but how is it that it is also the number of (ordered) subsets containing exactly j successes in the list of permutations (with repetition) of n binary elements (e.g. (3,2) = {SSF, SFS, FSS}) ?

Answer 1

I think I got it... I have to see the outcomes of the Bernoulli trials not as SFS, SFF, FSS, etc.. but as the list of the steps where successes happened: respectively (1,3), (1), (2,3), etc.. in this way (n,j) represents the subset of 2 successes in a n-steps experiment..