A $4$ by $4$ tic$-$tac$-$toe board is an $2-$dimensional array of char in which each element is either an $\text{'}$X$\text{'},\text{'}$O$\text{'}$ or $\text{'}-\text{'}.$ The game is like normal tic$-$tac$-$toe but the players may keep playing even after someone has three in a row. After the board is full, the winner is whichever player with the most "threes in a row".

a$)$ Calculate n $=$ the number of possible boards, without regard for the rules of tic$-$tac$-$toe. For example we'll include boards with $5$ X$\text{'}$s and $2$ O$\text{'}$s even though no such board could occur in an actual game.