The question asks us to determine the cations that can occupy specific positions in the structure of an inverse spinel \(AB_2O_4\). To solve this, we need to understand the structural characteristics and the ionic arrangement in an inverse spinel.

In a spinel structure, the formula is typically \(AB_2O_4\), where:

• \(A^{2+}\) is a divalent metal ion.
• \(B^{3+}\) is a trivalent metal ion.

For an inverse spinel, the \(A^{2+}\) ions occupy half of the octahedral sites while the \(B^{3+}\) ions are split between the remaining octahedral and all tetrahedral sites. Thus, an inverse spinel can be described as \((B^{3+})[A^{2+}B^{3+}]O_4\), with the square brackets denoting the octahedral sites.

Let's analyze the given options:

1. \(Ni(II)\) and \(Ga(III)\):
   • \(Ni^{2+}\) is divalent and can occupy octahedral sites.
   • \(Ga^{3+}\) is trivalent and can occupy octahedral and tetrahedral sites.
   • This matches the characteristics of an inverse spinel.
2. \(Zn(II)\) and \(Fe(III)\):
   • Though \(Fe(III)\) is suitable for the spinel structure, \(Zn(II)\) typically prefers to occupy tetrahedral sites, making it more likely for a normal spinel configuration.
3. \(Fe(II)\) and \(Cr(III)\):
   • \(Fe(II)\) and \(Cr(III)\) can exist in spinel structures but do not prefer an inverse arrangement due to other stabilization preferences.
4. \(Mn(II)\) and \(Mn(III)\):
   • Manganese ions can exist in both states but tend towards other potentially significant disproportionation reactions rather than perfectly fitting the inverse spinel criteria.

Thus, the correct answer for an inverse spinel configuration is: \(Ni(II)\) and \(Ga(III)\). This configuration allows \(Ni^{2+}\) to be in the octahedral sites and \(Ga^{3+}\) to distribute across both available site types, conforming with the inverse spinel structure.