I am trying to understand how to find binomial coefficients modulo a power of a prime. I am reading the paper by Andrew Granville for this. But I am unable to understand it completely. More specifically, I am unable to work out how each of $(N_j!)_p$ is computed efficiently. It would be really awesome if someone could show a small hand-worked example too - say $\binom{16}{5}$ mod $3^3$ or any other small example. Thanks in advance!
Edit Note: Earlier I had mentioned that I am unable to work out an example of the theorem by hand, but found out I was making a mistake. So have edited this question to understand how these coefficients are computed efficiently.