  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.
