The number of $3\times2$ matrices $A$, which can be formed using the elements of the set ${-2,-1,0,1,2}$ such that the sum of all the diagonal elements of $A^TA$ is $5$, is ______.

Solution: \( A = \begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \\ a_3 & b_3 \end{bmatrix} \) --- \( A^T A = \begin{bmatrix} a_1^2 + a_2^2 + a_3^2 & * \\ * & b_1^2 + b_2^2 + b_3^2 \end{bmatrix} \) --- Condition: \( a_1^2 + a_2^2 + a_3^2 + b_1^2 + b_2^2 + b_3^2 = 5 \) --- Count possible combinations --- Total ways \( = 312 \)