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The BK inequality (\cite{BK85}) says that,for product measures on$\{0,1\}^n$, the probability that two increasing events $A$ and $B$ `occurdisjointly' is at most the product of the two individual probabilities. Theconjecture in \cite{BK85} that this holds for {\em all} events was proved byReimer (cite{R00}).  Several other problems in this area remained open. For instance, although itis easy to see that non-product measures cannot satisfy the above inequalityfor {\em all} events,there are several such measures which, intuitively, shouldsatisfy the inequality for all{\em increasing} events. One of the most naturalcandidates is the measure assigning equal probabilities to all configurationswith exactly $k$ 1's (and probability 0 to all other configurations).  The main contribution of this paper is a proof for these measures. We alsopoint out how our result extends to weighted versions of these measures, and toproducts of such measures.

A BK inequality for randomly drawn subsets of fixed size