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# If four dice are rolled and the number of 1's, 2's, 3's, 4's, 5's, and 6's are recorded, how many possible results are there?

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If four dice are rolled and the number of 1's, 2's, 3's, 4's, 5's, and 6's are recorded, how many possible results are there?

This is confusing me a little bit. Naturally, I would say 6^4 = 1296 results. However, we could have 1123 and 1321 and still have the same results. How do I account for this overcounting?

Jun 13, 2022

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Your 6^4 ==1,296 is correct. That is called the "sample space". This 1,296 counts ALL the permutations possible in tossing 4 dice. The permutations will start at: 1111 and end in: 6666. The two examples you gave are already included in 1,296 permutations. They start like this:

1111 , 1112 , 1113 , 1114 , 1115 , 1116 , 1121 , 1122 , 1123 , 1124 , 1125 , 1126 , 1131 , 1132 , 1133 , 1134 , 1135.........  6651 , 6652 , 6653 , 6654 , 6655 , 6656 , 6661 , 6662 , 6663 , 6664 , 6665 , 6666 , Total =  1296 permutations.

Jun 13, 2022