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December 1st, 2009, 11:26 AM  #1 
Newbie Joined: Dec 2009 Posts: 1 Thanks: 0  Dice probability
I am trying to find a general solution to calculate a certain number with x number of dice. As in, what is the probability to roll a 10 with 3 6sided dice? Or to roll a 17 with 4 6sided dice? Could anyone please help me on this?

December 1st, 2009, 11:49 AM  #2 
Member Joined: Oct 2009 Posts: 64 Thanks: 0  Re: Dice probability
It's very easy to write the recursion relation. Let C(N,k) be the number of ways to roll k dice so that the total is N. Then C(N,k+1) = C(N1,k) + C(N2,k) + ... + C(N6,k) And C(N, 1) = 1 if 1<=N<=6 and 0 otherwise. That is, you look at the outcome of the first die, then count the number of ways to roll the remaining part of N on the remaining k dice. Add the counts for each possible roll of the first die, and you have the C(N,k+1) as desired. If you want to write a closed formula, you'll need to go a bit further. 
December 1st, 2009, 12:15 PM  #3 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  Re: Dice probability Wikipedia states that the probability of rolling k with n ssided dice is: So the probability of rolling 10 with 3 sixsided dice is . And the probability of rolling 17 with 4 sixsided dice is If you want to solve the problem for all possible k at once, you can look instead at the coefficients of the expansion of (x+x²+x³+...+x^s)^n. For example, if you tell Wolfram Alpha to expand (x+x^2+x^3+x^4+x^5+x^6)^3, which represents three 6sided dice, you get: which means that there is 1 way to get 18, 3 ways to get 17, 6 ways to get 16, 10 ways to get 15... etc. 

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