# Two six-sided dice are rolled and their product

Two six-sided dice are rolled and their product is taken. How many elements are in the subset of even numbers?
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Step 1
When two six-sided dice are rolled the sample space is as shown below
$\left(1,1\right),\left(1,2\right),\left(1,3\right),\left(1,4\right),\left(1,5\right)\left(1,6\right)$
$\left(2,1\right),\left(2,2\right),\left(2,3\right),\left(2,4\right),\left(2,5\right),\left(2,6\right)$
$\left(3,1\right),\left(3,2\right),\left(3,3\right),\left(3,4\right),\left(3,5\right),\left(3,6\right)$
$\left(4,1\right),\left(4,2\right),\left(4,3\right),\left(4,4\right),\left(4,5\right).\left(4,6\right)$
$\left(5,1\right).\left(5,2\right),\left(5,3\right),\left(5,4\right),\left(5,5\right).\left(5,6\right)$
$\left(6,1\right),\left(6,2\right),\left(6,3\right),\left(6,4\right),\left(6,5\right),\left(6,6\right)$
Step 2The subset of the elements where the product is even is shown below
$\left[\left(1,2\right),\left(1,4\right),\left(1,6\right),\left(2,1\right)\left(2,2\right)\left(2,3\right),\left(2,4\right)$
$\left(2,5\right),\left(2,6\right),\left(3,2\right),\left(3,4\right),\left(3,6\right),\left(4,1\right),\left(4,2\right)$
$\left(4,3\right),\left(4,4\right),\left(4,5\right),\left(4,6\right),\left(5,2\right),\left(5,4\right),\left(5,6\right)$
$\left(6,1\right),\left(6,2\right),\left(6,3\right),\left(6,4\right),\left(6,5\right).\left(6,6\right)\right]$
Therefore of elements in the subset of even numbers=27