Formula Blitz

For an array of non-negative integers $A$ of length $N$, define $f_{\text{and}}(A, k)$ to be the sum of the bitwise AND, of every (multi)subset of $k$ elements of $A$. Similarly define $f_{\text{or}}(A, k)$ and $f_{\text{xor}}(A, k)$. Formally, $$ f_{\text{and}}(A, k) = \sum_{\substack{S \subseteq A \\ |S| = k}} \text{AND}(S) $$ where $\text{AND}(S)$ denotes the bitwise AND of the elements of $