A \(\lambda\)-fold triple system is a pair \(T=(V,\text{QTR}{\mathcal B})\), \(\text{QTR}{\mathcal B}\) being a collection of 3-subsets of \(V,\) so that each \(2-\)subset of \(V\) occurs in exactly \(\lambda \) block of \(\text{QTR} {\mathcal B}\). If there is no \(\text{QTR}{\mathcal B}'\subset \text{QTR}{\mathcal B}\) so that \((V,\text{QTR}{\mathcal B}')\) is a \(\lambda '\)-fold triple system, \(\lambda '15\) is determined.