Number of independent components in a general four index tensor

Question

A general four-index tensor has $4^4=256$ indepedent components.

Why does something like $R_{abcd}$ mean $4^4$ components?

Why are these components independent (for a general tensor)?

What are these components exactly?

Answer 1

A rank-$(p,\,q)$ tensor in $n$-dimensional spacetime has $n^{p+q}$ entries, which in a completely general tensor can be chosen independently. The Riemann tensor of course doesn't have that many independent components (i.e. $n^4$) because of its symmetries. In fact, it only has $n^2(n^2-1)/12$ of them (e.g. $20$ when $n=4$).