# Number of rolls with m many n-sided dice

• mathboy
In summary, The number of different rolls with m many n-sided dice, if order is not important, can be calculated using the formula C(m+n-1, m), which is the number of ways to choose m numbers from m+n-1 numbers. This formula allows for repetitions and can be found in resources such as http://oregonstate.edu/~peterseb/mth232/232euler_comb.html or http://www.csee.umbc.edu/~stephens/203/PDF/6-5.pdf. However, this formula may not work for larger values if the number of pips on a die cannot exceed n.

#### mathboy

[SOLVED] Number of rolls with m many n-sided dice

How many different rolls are there with m many n-sided dice, if order is not important?

If order was important, the answer would simply be n^m. But if order is not important?

If two 6-sided dice are rolled, there are C(6,2)+C(6,1) = 21 different rolls.
If three 6-sided dice are rolled, there are C(6,3)+C(6,1)C(5,1)+C(6,1) = 56 different rolls.

Is there a general method for rolling m many n-sided dice? Or do you have to continue adding up all the cases? Perhaps a general method involving a matrix with m indices (e.g. a tensor) and simply excluding all permutations of the indices?

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Wouldn't it be n^m/m!? For example, say you have a black die and a red die, both 6-sided. The number of possible rolls is 6^2. If you ignore the color of the dice, then you have to divide by 2.

No. The number of rolls with 2 six-sided dice (if order is unimportant) is 21, not 18. Rolls with doubles are not being counted twice if order is considered important.

Oh that's right. Sorry about that. I don't know of a general method. I'm thinking that you would first have to count the number of rolls where all dice show the same side, then for m-1 dice showing the same side, then for m-2 dice, etc.

That's exactly what I did in my opening post. But it gets awkward after a while (e.g. for 7 dice, you can have 2 doubles and 1 triple). I'm looking for a better method that gives a single formula for arbitrarily values of m and n,

e.g. m^n minus something.

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mathboy said:
How many different rolls are there with m many n-sided dice, if order is not important?

If order was important, the answer would simply be n^m. But if order is not important?

If two 6-sided dice are rolled, there are C(6,2)+C(6,1) = 21 different rolls.
If three 6-sided dice are rolled, there are C(6,3)+C(6,1)C(5,1)+C(6,1) = 56 different rolls.

Is there a general method for rolling m many n-sided dice? Or do you have to continue adding up all the cases? Perhaps a general method involving a matrix with m indices (e.g. a tensor) and simply excluding all permutations of the indices?

Yes.

$$\frac{(n + m - 1)!}{m!(n - 1)!}$$

CS

Wow! That gives the right answer for all the cases I worked out.

So the answer is C(m+n-1,m), i.e. the number of ways to choose m numbers from m+n-1 numbers. Half of this makes some sense to me because we are rolling m dice. But what's the reason for the m+n-1?

mathboy said:
Wow! That gives the right answer for all the cases I worked out.

So the answer is C(m+n-1,m), i.e. the number of ways to choose m numbers from m+n-1 numbers. Half of this makes some sense to me because we are rolling m dice. But what's the reason for the m+n-1?

It's just a combination that allows repetitions.

http://oregonstate.edu/~peterseb/mth232/232euler_comb.html [Broken]

or

http://www.csee.umbc.edu/~stephens/203/PDF/6-5.pdf

Hope that helps.

CS

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Thank you so much. Sometimes extra knowledge is the key to solving a problem.

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## What is the formula for calculating the number of rolls with m many n-sided dice?

The formula for calculating the number of rolls with m many n-sided dice is nm. This means that for every additional die, the number of possible outcomes is multiplied by the number of sides on the die.

## How does the number of sides on the dice affect the number of possible rolls?

The number of sides on the dice directly affects the number of possible rolls. The more sides on the dice, the higher the number of possible outcomes, and therefore the higher the number of rolls that can be made with a given number of dice.

## What is the difference between the number of rolls and the number of possible outcomes?

The number of rolls refers to the actual number of times the dice are rolled, while the number of possible outcomes refers to the total number of outcomes that could occur from those rolls. For example, rolling 2 six-sided dice (2d6) has 36 possible outcomes, but only 11 possible rolls (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12).

## How does the number of dice affect the probability of getting a specific roll?

The number of dice does not affect the probability of getting a specific roll. The probability of getting a specific roll is based on the number of possible outcomes and is the same regardless of the number of dice being rolled.

## Can the number of rolls with m many n-sided dice be used to determine the odds of winning a game?

The number of rolls with m many n-sided dice can be used to determine the odds of winning a game if the game involves rolling dice. However, other factors such as the rules of the game and the strategy used by the players also play a role in determining the odds of winning.