How Can You Calculate the Probability of Rolling a Sum of 14 with 7 Dice?

[leumas614]

Homework Statement

Roll 7 dice. What is the probability that the sum of the numbers is 14?

Homework Equations

Possibly a probability generating function because the problem comes from the chapter that deals with it but I can't think of which one (if any actually apply).

The Attempt at a Solution

Here is the solution: 6^-7[(13 choose 7)-49]

I really have no idea why. The 6⁷ in the denominator I understand because that is the total number of different ways you can roll 7 dice. Where does the rest come from?

 

[yyat]

By the definition of probability, the numerator should be the number of different ways you can roll seven dice and get a sum of 14. So how would you count them to get this answer?

 

[Mark44]

You might start with a simpler problem to get your head around the harder problem. For example, what's the probability of rolling three dice and getting a total of 6?

The only combinations that produce a total of 6 are (2, 2, 2) and (1, 2, 3). There is only one way to get (2, 2, 2), namely that all three dice have to be 3s, but there are 6 ways to get (1, 2, 3). I've written this as an ordered triple, but I really don't mean it that way--only that one die is a one, another is a two, and the last is a three.

So for this easier problem, what's the probability that three dice will show a total of 6? Can you extend this idea to your problem?



Can you extend this counting strategy to