Question 11
Define a *T-grid* to be a $3 \times 3$ matrix which satisfies the following two properties:
$(1)$ Exactly five of the entries are $1$’s, and the remaining four entries are $0$’s.
$(2)$ 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*.
$(1)$ Exactly five of the entries are $1$’s, and the remaining four entries are $0$’s.
$(2)$ 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*.