Dice Permutation Calculator: How It Works

This calculator works out the number of distinct permutations you can roll with a set of dice, given how many dice you have and how many sides each die has. You enter the number of dice and the number of sides per die, and it returns the total count of possible ordered outcomes (permutations), along with the number of combinations if order doesn’t matter. The result is the exact count of different sequences you could roll, such as 36 for two six-sided dice.

Formula

The core formula for permutations with repetition is:

P = sⁿ

• P = number of permutations (ordered outcomes)
• s = number of sides on each die (e.g., 6 for a standard die)
• n = number of dice rolled

This works because each die is independent: the first die has s possible results, the second also has s, and so on. Multiplying them gives s × s × … × s (n times) = sⁿ.

For combinations (where order doesn’t matter, e.g., rolling a 2 and a 5 is the same as a 5 and a 2), the formula is the stars and bars method:

C = C(s + n – 1, n) = (s + n – 1)! / (n! × (s – 1)!)

Here, C(a, b) is the binomial coefficient (combinations of a items taken b at a time). This counts the number of multisets of size n from s categories.

Worked Example

Using the default values: 2 dice, 6 sides each.

Permutations: P = 6² = 6 × 6 = 36

Combinations: C(6 + 2 – 1, 2) = C(7, 2) = 7! / (2! × 5!) = (7 × 6) / (2 × 1) = 42 / 2 = 21

So with two six-sided dice, there are 36 ordered outcomes (like (1,2) vs (2,1)) and 21 unordered outcomes (where (1,2) and (2,1) count as the same).

When to Use This Calculator

You might need dice permutations for:

• Game design: Determining how many distinct outcomes a dice mechanic produces, e.g., for a custom dice pool system.
• Probability calculations: The total number of outcomes is the denominator when computing probabilities of specific rolls.
• Educational exercises: Teaching combinatorics or probability with concrete examples.
• Simulation planning: Estimating the state space for a Monte Carlo simulation of dice rolls.

For instance, if you’re designing a game where you roll 3d6 and add the results, the total number of permutations is 6³ = 216. That’s the denominator for any probability like “rolling a total of 10” (which has 27 permutations, so probability = 27/216 = 12.5%).

How to Read the Results

The permutations number tells you how many distinct sequences of dice faces are possible. Each sequence is considered different even if the same faces appear in a different order. For example, with two dice, (1,2) and (2,1) are two different permutations.

The combinations number (shown as a secondary output) treats rolls with the same multiset of faces as identical. So (1,2) and (2,1) count as one combination. This is useful when you only care about the set of faces, not the order they appear in.

Both numbers grow quickly. For 3d6, permutations = 216, combinations = C(8,3) = 56. For 5d6, permutations = 7776, combinations = C(10,5) = 252. The gap widens as n increases.

Common Mistakes and Misconceptions

A frequent error is confusing permutations with combinations. If you’re calculating the probability of rolling a specific total (like 10 on 3d6), you need permutations because each ordered sequence is equally likely. But if you’re counting distinct outcomes for a game where order doesn’t matter (e.g., you just look at the faces rolled), you need combinations.

Another mistake is using the wrong base. If you have dice with different numbers of sides (e.g., one d6 and one d8), the formula becomes s1 × s2, not sⁿ. This calculator assumes all dice have the same number of sides. For mixed dice, multiply the side counts together.

Also, don’t confuse permutations with arrangements of a subset. Here we’re counting all possible outcomes of rolling n dice, not arranging a subset of faces.

Limitations and Extensions

This calculator handles the basic case of identical dice with the same number of sides. It does not account for:

• Exploding dice (where a max roll rerolls and adds): the outcome count becomes infinite in theory, so you’d need a different approach.
• Dice pools with target numbers (e.g., count successes): you’d need to sum over binomial distributions.
• Weighted dice (non-uniform probabilities): the total count is still sⁿ, but the probability distribution changes.

For those cases, you’d need a more specialized calculator or a simulation. The formulas here assume fair, independent dice.

Frequently Asked Questions

How many permutations are there for 3d6?
For 3 six-sided dice, the number of ordered outcomes is 6³ = 216. That’s the total number of distinct sequences you can roll, from (1,1,1) to (6,6,6).

What’s the difference between permutations and combinations for dice?
Permutations count every ordered sequence as distinct: (1,2) and (2,1) are different. Combinations treat them as the same because they contain the same faces. For 2d6, permutations = 36, combinations = 21.

Does this work for dice with different numbers of sides?
No, the calculator assumes all dice have the same number of sides. If you have, say, one d6 and one d8, the total permutations are 6 × 8 = 48, not 6² or 8².

Why does the number of combinations use the stars and bars formula?
Because combinations of dice rolls are multisets: you’re choosing n items from s categories with repetition allowed, and order doesn’t matter. The stars and bars formula C(s+n-1, n) counts exactly that.

How do I calculate the probability of a specific total?
First, find the number of permutations that give that total. For example, with 2d6, the total 7 has 6 permutations: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Then divide by the total permutations (36) to get 6/36 = 1/6 ≈ 16.67%.

What if I roll more than 10 dice?
The calculator handles any positive integer for the number of dice, but the numbers get astronomically large quickly. For 10d6, permutations = 6¹⁰ = 60,466,176. The calculator can display that, but for larger numbers you might need scientific notation.

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in n | Number of dice | dice | 2 | 1
in s | Sides per die | sides | 6 | 2
out perm | Permutations (ordered) | - | s^n | 0
out comb | Combinations (unordered) | - | ncr(s+n-1, n) | 0