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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


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