Školjka

%V0 Let $n$ be a positive integer. Show that the numbers $$\binom{2^n - 1}{0},\; \binom{2^n - 1}{1},\; \binom{2^n - 1}{2},\; \ldots,\; \binom{2^n - 1}{2^{n - 1} - 1}$$ are congruent modulo $2^n$ to $1$, $3$, $5$, $\ldots$, $2^n - 1$ in some order. Proposed by Duskan Dukic, Serbia