  A tree $T$, in an edge-colored graph $G$, is called {\em a rainbow tree} if
no two edges of $T$ are assigned the same color. For a vertex subset $S\in
V(G)$, a tree that connects $S$ in $G$ is called an $S$-tree. A {\em
$k$-rainbow coloring} of $G$ is an edge coloring of $G$ having the property
that for every set $S$ of $k$ vertices of $G$, there exists a rainbow $S$-tree
$T$ in $G$. The minimum number of colors needed in a $k$-rainbow coloring of
$G$ is the {\em $k$-rainbow index of $G$}, denoted by $rx_k(G)$. In this paper,
we obtain the exact values of $rx_3(K_{2,t})$ for any $t\geq 1$.
