# Apply formula on random sampling without replacement, but with replacement between each iterations

I am looking for an algebraic solution to explain that I apply a formula on a vector constituted of a random sampling of n elements in a population of size N without replacement.

The formula is applied B times (bootstraped iteration) with replacement in original populations between each iterations.

If I understood well, I can write that inside each iteration, probability for an element to be sampled P(e) is:

$P(e)&space;=&space;\frac{(N-n)!}{N!}$

And probability of the whole vector S of size n to be sampled P(s) is:

$P(S)&space;=&space;\frac{1}{\binom{N}{n}}$

But how to explain that between each iteration B, probabilities P(e) and P(S) are restaured to their origin ?

Is it OK to say that $P(e)&space;=&space;\frac{(N-n)!}{N!}$ and $P(S)&space;=&space;\frac{1}{\binom{N}{n}}$ inside an iteration; and that $P(e)&space;=&space;\frac{1}{N}$ between iteration? And what about P(S) ?