  Let $G$ be a finite additive abelian group with exponent $n$ and
$S=g_{1}\cdots g_{t}$ be a sequence of elements in $G$. For any element $g$ of
$G$ and $A\subseteq\{1,2,\ldots,n-1\}$, let $N_{A,g}(S)$ denote the number of
subsequences $T=\prod_{i\in I}g_{i}$ of $S$ such that $\sum_{i\in
I}a_{i}g_{i}=g$ , where $I\subseteq\left\{ 1,\ldots,t\right\} $ and $a_{i}\in
A$. In this paper, we prove that $N_{A,0}(S)\geq2^{|S|-D_{A}(G)+1}$, when
$A=\left\{ 1,\ldots,n-1\right\} $, where $D_{A}(G)$ is the smallest positive
integer $l$, such that every sequence $S$ over $G$ of length at least $l$ has
nonempty subsequence $T=\prod_{i\in I}g_{i}$ such that $\sum_{i\in
I}a_{i}g_{i}=0$, $I\subseteq\left\{ 1,\ldots,t\right\} $ and $a_{i}\in A$.
Moreover, we classify the sequences such that $N_{A,0}(S)=2^{|S|-D_{A}(G)+1}$,
where the exponent of $G$ is an odd number.
