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Note all valid subsets one-to-one correspond to all permutations of unlabelled white balls and k unlabelled black balls where no two black balls are adjacent. Also, each such permutation can be obtained by inserting the k black balls into the gaps between two white balls (including the leftmost and the rightmost positions) such that no two black balls are inserted into the same gap.
There are gaps, so there are ways to insert black balls.
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