Ne'er give up!

How many integer solutions are there to the equation

$\sum _{ i=1 }^{ 12 }{ { x }_{ i } } \quad =\quad 264$

$(i)$ with ${x}_{i} \geq 0$ ?

$(ii)$ with ${x}_{1}, {x}_{2}, {x}_{3}......{x}_{7} \geq 2$ , ${x}_{8}, {x}_{9} \geq 6$ and ${x}_{10}, {x}_{11}, {x}_{12} \geq 4$ ?

Step 1 Add the answers of $(i), (ii)$ and multiply by $39916800$ .

Step 2 Now, change the $\times$ sign with $+$ sign.(Don't change any other operator, this change has to be done when the term is without any fraction)

Step 3 Do the calculations and then give your answer.

Extra Credit - Why I chose the number 264? :P