For every finite Abelian group Γ and for all $g,a₁,...,a_k\in \Gamma$, if there exists a solution of the equation ${\sum }_{i=1}^k{a}_i{x}_i=g$ in non-negative integers $x_i\le b_i$, where $b_i$ are positive integers, then the number of such solutions is estimated from below in the best possible way.