Any connection between 3 concepts and probability

[davedave]

I have posted a similar problem not long ago.This time, I am trying to find a connection between 3 concepts that would possibly lead to the answer to the problem.

I have heard that there is a way to find the probability of getting a particular sum by rolling multiple six-sided dice at the same time without listing ALL the outcomes which is really horrifying.

Someone told me that there is a connection among polynomials, Pascal's triangle, combinations and dice.

For example, say you roll 5 six-sided fair dice and want to find the probability of getting a sum of 17. In this case, there are 780 combinations of such a sum and 6^5 different sums.
So, the probability of the sum of 17 is 0.100308641975309.

That person said that you consider the polynomial expansion of
(1+x+x^2+x^3+x^4+x^5)^5 .

Each term is raised to ascending powers. The 6 terms in there represent the 6 sides of each die and the 5th power represents the 5 dice. If you can connect it to Pascal's triangle and combinations, you will get the correct answer as stated above.

I am not sure if I wrote down the polynomial correctly. I lost contact with that person.

Can someone tell me how it is possible to solve the 5 six-sided dice problem with those concepts? Thanks.

 

[bpet]

... you consider the polynomial expansion of
(1+x+x^2+x^3+x^4+x^5)^5

It might be easier to work with (x+x^2+...+x^6)^5. That way the exponent of x corresponds to the dice score and the coefficient is the number of possible ways of getting that score. If you're not sure, try the examples with 1 dice and 2 dice by hand to convince yourself that it works. Does that help?

 

[csopi]

You should take a look on the concept of "probability-generating function".