5 People roll a dice and flip a coin: number of possible outcomes 
Each of 5 people flip a coin and roll a dice (six sides). Find the number of possible combinations.

Progress
I know the total number of possibilities equates to $6 \times 2$ because the dice has 6 options, and the coin has 2 options. As a result we have 12 different options for one instance of this.
However 5 people are doing this. So our number of total combinations increases greatly!
As a result, I am most confident that the number of combinations is now $12^5$, however, I am unsure of how to find "specific" results such as though asked below. Any guidance into a direction would be greatly appreciated.

*

*How many outcomes are in the event where nobody rolls a six?


*How many outcomes are in the event where at least one person rolls a six?
 A: ** How many outcomes are in the event where nobody rolls a six?
   If they can't roll a six, there are 5 other numbers to roll, and either coin-flip is still allowed. So each person has $2\times 5=10$ possible outcomes. Since there are 5 people, there are $10^5$ possible outcomes.
** How many outcomes are in the event where at least one person rolls a six?
   Well, either there are no sixes rolled or at least one person rolls a six, so this result is the difference of your answer and the answer above $12^5 - 10^5$.
A: Right, so far.  
The total number of outcomes, our total probability space, is $(6\times 2)^5$, because each of 5 people can independently have $6\times 2$ results; so we multiply that five times (ie: take it to the 5th power).
$$|U|=(6\times 2)^5 = 12^5$$
Now if we want the first favoured space of nobody rolls a six, the we reduce the outcomes each person can get to $5$ dice results and $2$ coin results.  Use the same principle to complete:
$$|S_{(n_6=0)}| = \ldots\ldots\ldots$$
For the second favoured space, of at least one rolls a six, we must note that this is the complement of nobody rolling a six.   So the count is the total minus the complement:
$$\begin{align}|S_{(n_6\geq 1)}| & = |U|-|S_{(n_6=0)}| \\ & = \ldots\ldots\ldots\end{align}$$
