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Let it be a tetrahedron with the numbers $1$,$2$,$3$ and $4$ on its faces.The tetrahedron is launch $3$ times. Each time, the number that stays face down is registered.

$1$)In total how many possible ways there are to registered the $3$ launches?

As there are $4$ numbers to $3$ launches(positions), the order matters and each number can repeats itself. I used a permutation with replacement: $4^3=64$

$2$)How many possible ways there are to the number $1$ never face down?

In this case I reduced the sample set to $\{2,3,4 \}$, and made the same as before. But this time there are $3$ numbers to $3$ launchs: $3^3=27$

$3$)How many possible ways there are to the number $1$ appears only $1$ time face down?

There are $3$ ways for number $1$ can be put on the $3$ launches. For the $2$ left there is $\{2,3,4 \}$.So I made a permutation with replacement: $3 \cdot 3^2=27$

$4$)How many possible ways there are to the number $1$ appears exactly $2$ times face down?

First I made a combination: $C(3,2)$ to find the number of ways that the pair of $1$'s can be put in the $3$ launchs.Then I multiplied by $3$, that is $ \{2,3,4 \}$ : $C(3,2) \cdot 3=9$.

Is this correct? Thank you very much, you(plural)have been very helpful.

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    In (3), you mean $3 \cdot 3^2=27$.2012-01-20
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    There is also $1$ way the number $1$ appears three times face down, and the fact that $27+27+9+1=64$ should be encouraging.2012-01-20
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    That sounds right. As a check, you should argue that $1$ can appear on the bottom on no rolls, one roll, two rolls, or three rolls (there are no other possibilities), and you have worked out the number of ways that $1$ never appears, appears once, and appears twice for a total of $27 + 27 + 9 = 63$ of the $64$ occurrences. The remaining possibility is that of $1$ on all three rolls, right? Of course, all the numbers adding up doesn't _guarantee_ that they are correct, but if they _don't_ add up, that suggests that the work and the argument needs to be checked.2012-01-20
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    @Dilip You should make that an answer.2012-01-20
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    The work was nicely done.2012-01-20
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    João, on this site, people are allowed, in fact encouraged, to post their own answers to problems, and to accept them too. So why don't you post your work as an answer (after a little clean-up of the write-up you already have, and possibly incorporating the suggestions for a check that Henry and I suggested)? As André Nicolas and Thanassis said, the work was nicely done.2012-01-21

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