  Given a family of graphs $\mathcal{F}$, the Tur\'{a}n number $ex(n,
\mathcal{F})$ denotes the maximum number of edges in any $\mathcal{F}$-free
graph on $n$ vertices. Recently, Alon and Frankl studied of maximum number of
edges in an $n$-vertex $\{K_{k+1}, M_{s+1}\}$-free graph, where $K_{k+1}$ is a
complete graph on $k+1$ vertices and $M_{s+1}$ is a matching of $s+1$ edges.
They determined the exact value of $ex(n, \{K_{k+1},M_{s+1}\})$. In this paper,
we extend the matching $M_{s+1}$ to star forest $(s+1)S_l$, and determine the
exact value of $ex(n, \{K_{k+1},(s+1)S_l\})$ for sufficiently large enough $n$.
Furthermore, all the extremal graphs are obtained.
