 Solution

Distinguish the $N$ balls by labels. For a fixed position $n$, each label appears there in exactly $(N-1)!$ of the $N!$ equally likely permutations. Since $N_B$ labels are black,This is the same marginal law at every position, even the last one. The argument averages over earlier draw outcomes; it does not assert that a conditional probability after a specified history remains $N_B/N$. The original PDF uses $B_n$, correcting the transcription's uppercase subscript $B_N$.