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There is one unspecified thing about this problem. As said above, --1 --2 3-6-4 and --5 --6 --2 4-1 3 --5 Look like the same die. So, in reality, there are two answers to this problem- one in which the above to are the same, and one in which the above is a discrepancy ( for example, each side of the die is a different color and youre asking how many differences there are in that possibility. In these situations, i find it best to stay away from factorials and random functions you learned in middle school that seem to always work but actually dont in most situations. Just take each space- the number of possibilities for each, and multiply. For case number two, in which the above examples are counted as different, the answer is 48. How? Ok, so Cube. O's will represent open spaces and X's will represent Closed ones (while the others represent the number of possibilities) --O --O .OOO --O The topmost space has six possibilities, so we then count --6 --O .OXO --O The center is an X because once one of the 6 numbers is chosen to be at the top, then there must be a certain number in the middle there. Afterwards we get --6 --4 .OXO --X 4 possibilities for the second number for four numbers remained. The final looks like --6 --4 .2XX --X The number of possibilities is 6x4x2, which is 48. However, if the difference stated above doesnt matter, then there are actually only 6 possibilities. (or at least im pretty sure, although others said two). I think its three because, well, think about it. Just think about the numbers 1, 2, and 3. On this die that is mentioned, the numbers 1,2,3 must ALL be present/touching/3d visible at one and only one vertice(sp?) of the cube (like, all must be present at some corner here http://home.cc.umanitoba.ca/~gunderso/model_photos/platonic/cube.jpg where one two and three are all seen). This must be true because on a cube the only two numbers that dont touch are ones opposite each other, and neither 1 nor 2 nor 3 is opposite to 1 , 2, or 3. Thus the only difference is where the 1, 2, and 3 are relative to eachother. 3 numbers, 3 places, 3!. 6

There is one unspecified thing about this problem. As said above, --1 --2 3-6-4 and --5 --6 --2 4-1 3 --5 Look like the same die. So, in reality, there are two answers to this problem- one in which the above to are the same, and one in which the above is a discrepancy ( for example, each side of the die is a different color and youre asking how many differences there are in that possibility. In these situations, i find it best to stay away from factorials and random functions you learned in middle school that seem to always work but actually dont in most situations. Just take each space- the number of possibilities for each, and multiply. For case number two, in which the above examples are counted as different, the answer is 48. How? Ok, so Cube. O's will represent open spaces and X's will represent Closed ones (while the others represent the number of possibilities) --O --O .OOO --O The topmost space has six possibilities, so we then count --6 --O .OXO --O The center is an X because once one of the 6 numbers is chosen to be at the top, then there must be a certain number in the middle there. Afterwards we get --6 --4 .OXO --X 4 possibilities for the second number for four numbers remained. The final looks like --6 --4 .2XX --X The number of possibilities is 6x4x2, which is 48. However, if the difference stated above doesnt matter, then there are actually only 6 possibilities. If you don't believe me then i can draw it out, but the way i did it cant really be done with ascii.