The packing number \(D(2,k,v)\) is defined to be the maximum cardinality of a family of \(k\)-subsets, chosen from a set of \(v\) elements, such that no pair of elements appears in more than one \(k\)-subset. We examine \(D(2,k,v)\) for \(v < k(k-1)\) and determine such numbers for the case \(k=5\), \(v < 20\).