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Let G be a finite additive abelian group with exponent n and S = g 1 ⋯ g t be a sequence of elements in G. For any element g of G and A ⊆ { 1 , 2 , ... , n − 1 } , let N A , g (S) denote the number of subsequences T = ∏ i ∈ I g i of S such that ∑ i ∈ I a i g i = g , where I ⊆ { 1 , ... , t } and a i ∈ A. In this paper, we prove that N A , 0 (S) ≥ 2 | S | − D A (G) + 1 , when A = { 1 , ... , n − 1 } , 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 = ∏ i ∈ I g i such that ∑ i ∈ I a i g i = 0 , I ⊆ { 1 , ... , t } and a i ∈ 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. [ABSTRACT FROM AUTHOR]