Consider the set A of all determinants of order 3 with entries 0 or 1 only. Let B be the subset of A consisting of all determinants with value 1. Let C be the subset of the set of all determinants with value -1, then

B has same number of elements as C

$$\begin{gathered} \text{ It follows that }\mathrm{n}\left(\mathrm{C}\right)⩽\mathrm{n}\left(\mathrm{B}\right) \\ \text{ That is, the number of elements in B is less than or equal to the number of element in }\mathrm{C}\text{. } \\ \text{ Similarly, }n\left(C\right)⩽n\left(B\right)\text{. } \\ \text{ Hence }\mathrm{n}\left(\mathrm{B}\right)=\mathrm{n}\left(\mathrm{C}\right)\text{, That is }\mathrm{B}\text{ has as many element as }\mathrm{C}\text{. } \end{gathered}$$