Number of ways of making a die using the digits $1,2,3,4,5,6$ 
Find the number of ways of making a die using the digits $1,2,3,4,5,6$.

I know that $6!$ is not the correct answer because some arrangements can be obtained just by rotation of the dice. So there will be many repetitions. I tried by fixing any two opposite faces and using circular permutations for the remaining 4 faces. Number of such arrangements is $2!\binom{6}{2}(4-1)!=2\times15\times 6=180$.
But answer given is just $30$. Maybe there are still some repetitions which I am not seeing.
 A: fix one number on one face, then choose any of the 5 remaining numbers to go on the opposite face. the remaining 4 numbers can be arranged in a ring in $3!=6$ distinguishable ways, hence the answer is 30
A: Here's yet another way of thinking: $6!$ arrangements divided by the $4!=24$ rotational symmetries of the cube, so there are $5 \cdot 6=30$ ways.
A: Fix $1$ (say) at bottom, and any of $5$ others at the top, in $5$ ways.
Now remember that you can rotate the die w/o disturbing the top and bottom to show $4$ different faces, so the remaining numbers can be fixed in $4!/4 = 6$ ways
$5 \times 6 = 30$ 
A: The faces on a die are analogous to the vertices on an octahedron. You are correct in beginning with $6!.$ Now, we see how many times we have overcounted. Notice that for a fixed top vertex of the octahedron, which can be labeled as any of the $6$ numbers, there are four rotational congruencies. And as I have just mentioned, any of the six faces could be positioned as the top vertex. So each arrangement was counted $6 \times 4 = 24$ times. Accounting for this, the number of arrangements is $\frac{6!}{24} = \frac{720}{24} = \boxed{30}.$
A: Hint if you fix an arrangement then it has $6$ similar arrangements you see why by rotating in $2$ directions X,Z so answer should be $\frac{180}{6}=30$
