2022 AMC 10B Problem 18

Consider systems of three linear equations with unknowns x, y, and z, \begin{aligned} a_1 x + b_1 y + c_1 z & = 0 \\ a_2 x + b_2 y + c_2 z & = 0 \\ a_3 x + b_3 y + c_3 z & = 0 \end{aligned} where each of the coefficients is either 0 or 1 and the system has a solution other than x=y=z=0. For example, one such system is \{ 1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0 \} with a nonzero solution of \{x,y,z\} = \{1, -1, 1\}. How many such systems of equations are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.)

\textbf{(A)}\ 302 \qquad\textbf{(B)}\ 338 \qquad\textbf{(C)}\ 340 \qquad\textbf{(D)}\ 343 \qquad\textbf{(E)}\ 344

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.