Assume we have a set of \(k\) colors and we assign an arbitrary subset of these colors to each vertex of a graph \(G\). If we require that each vertex to which an empty set is assigned has in its neighborhood all \(k\) colors, then this assignment is called the \(k\)-rainbow dominating function of a graph \(G\). The minimum sum of numbers of assigned colors over all vertices of \(G\), denoted as \(\gamma_{rk}(G)\), is called the \(k\)-rainbow domination number of \(G\). In this paper, we prove that \(\gamma_{r2}(P(n, 3)) \geq \left\lceil \frac{7n}{8} \right\rceil.\)