Define a T-grid to be a $3\times3$ matrix which satisfies the following two properties: Exactly five of the entries are $1$'s, and the remaining four entries are $0$'s. Among the eight rows, columns, and long diagonals (the long diagonals are $\{a_{13},a_{22},a_{31}\}$ and $\{a_{11},a_{22},a_{33}\}$, no more than one of the eight has all three entries equal. Find the number of distinct T-grids.