Let {D₁, D₂, D₃, ..., D_n} be the set of third-order determinants that can be made with the distinct non-zero real numbers a₁, a₂, ..., a₉. Then 

The total number of third-order determinants is 9!. The number of determinants is even and of these there are 9!/2 pairs of determinants which are obtained by changing two consecutive rows, so $\sum _{\mathrm{i}=1}^{\mathrm{n}} {\mathrm{D}}_{\mathrm{i}}=0$.