Distinct matrices of size $n\times n$ such that every row and column contains at least one $'a'$
With 4 a's we will have $6\cdot \binom61 \cdot(25^5)$
This is not correct. It excludes some matrices where no 3 a's satisfy the conditions.
\begin{matrix}a&a&?\\?&?&a\\?&?&a\end{matrix}
With 5 a's we will have $6\cdot \binom62 \cdot(25^4)$
This is also incorrect. Some matrices are excluded:
\begin{matrix}a&?&?\\a&?&?\\a&a&a\end{matrix}
while others are counted twice:
\begin{matrix}a&\color{blue}a&?&&\color{blue}a&a&?\\\color{blue}a&a&?&=&a&\color{blue}a&?\\?&?&a&&?&?&a\end{matrix}