For a pair of positive integers $n,k$ with $n\geq 2$, in this paper we prove that $$ \sum_{ r=1}^k\sum_{ |\bf\alpha|=k}{ k\choose\bf\alpha} \zeta(n\bf\alpha)=\zeta(n)^k =\sum^k_{ r=1}\sum_{ |\bf\alpha|=k} { k\choose\bf\alpha}(-1)^{ k-r}\zeta^\star(n\bf\alpha), $$ where $\bf\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_r)$ is a $r$-tuple of positive integers. Moreover, we give an application to combinatorics and get the following identity: $$ \sum^{ 2k}_{ r=1}r!{ 2k\brace r}=\sum^k_{ p=1}\sum^k_{ q=1}{ k\brace p}{ k\brace q} p!q!D(p,q), $$ where ${ k\brace p}$ is the Stirling numbers of the second kind and $D(p,q)$ is the Delannoy number.
Categories: math.NT