Calculating odds of rolling a given total with a specified number of dice

[Dave Britten]

(06-15-2015 06:59 PM)Thomas Klemm Wrote:  

(06-15-2015 03:25 PM)Dave Britten Wrote:  For example, what are the odds of rolling a total of 22 on 9 dice? (It's about 1.45% in this case.)

It's the coefficient of \(x^{22}\) in:

\(
(\frac{x^6}{6}+\frac{x^5}{6}+\frac{x^4}{6}+\frac{x^3}{6}+\frac{x^2}{6}+\frac{x}{​6})^9
\)

The exact value is:

\(\frac{16211}{1119744}\)

Cheers
Thomas

That makes sense. What's the practical way to do a multinomial expansion like that, aside from feeding it through a CAS and letting it sum like terms? Seems like you'd have to find all permutations of 9 integers between 1 and 6 that add up to, e.g., 22, then you're back to square one.