  In this paper, we show that if a graph $G$ satisfies
$c_{1}(G-X)+\frac{2}{3}c_{3}(G-X)\leq \frac{4}{3}|X|+\frac{1}{3}$ for all
$X\subseteq V(G)$, then $G$ has a $\{P_{2},P_{5}\}$-factor, where $c_{i}(G-X)$
is the number of components $C$ of $G-X$ with $|V(C)|=i$.
