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#2587
Inverted Dice
Keymaster

Well, there are 120 possible rolls where the dice show no [1]’s, and [2] [3] [4] [5] [6] is just one of them. Another is [6] [3] [4] [5] [2] … and so on.

The first die can show [2], [3], [4], [5] or [6] (i.e. 5 possible values). Let’s say it shows [2], then the second die can show [3], [4], [5] or [6] (i.e. 4 possible values). Let’s say it shows [3], then the third die can show [4], [5] or [6] (i.e. 3 possible values). Let’s say it shows [4], then the fourth die can show [5] or [6] (i.e. 2 possible values). Let’s say it shows [5], then the fifth die must show [6] (i.e. 1 possible value) in order to achieve the inverted sum 1.

5 times 4 times 3 times 2 times 1 is 120, hence the probability 120/7776 (where 7776 is number of ways in which five dice can land).

When it comes to [1] [1] [1] [1] [1], there is really no other way, it can only be rolled in one way.
1 times 1 times 1 times 1 times 1 is 1, hence the probability 1/7776.