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We examine the well-known problem of determining the capacity of multidimensional run-length-limited constrained systems. By recasting the problem, which is essentially a combinatorial counting problem, into a probabilistic setting, we are able to derive new lower and upper bounds on the capacity of (0, k)-RLL systems. These bounds are better than all previously-known analytical bounds for k ≥ 2, and are tight asymptotically. Thus, we settle the open question: what is the rate at which the capacity of (0, k)-RLL systems converges to 1 as k → ∞? We also provide the first nontrivial upper bound on the capacity of general (d, k)-RLL systems.