[DF] Suitable Dangerous Encounters

[thulben]

Originally Posted by tjbuege

For example, say you inflict damage of 10d+2 (that's 12-62 points damage) vs a DR of 25. What are your chances of rolling penetrating damage? By my calculations there are 60,466,176 possible combinations of 10 dice (6^10). That's beyond my patience of figuring it out the long way! :)

The short answer to the question of "how many ways can I come up with the sum of n on 10 dice" is (x+x^2+x^3+x^4+x^5+x^6)^10, using the coefficient of x^n in the expanded polynomial

The long answer is that you essentially use the algebra of multiplying polynomials to do the bookkeeping. A normal 6-sided die has each number 1-6 on it. We need some way to keep track of every permutation. By using the exponent of a garbage variable as the value shown on the die, algebra gives us a simple method to do this. As you can see from the polynomial above, each number 1-6 is represented once. We then multiply it by itself 10 times to see what the effect of rolling the die 10 times would be. The nice thing about this method is that you can account for "non-standard" dice (i.e. the possible sums of rolling a 6-sided die with a 1 1's, 2 3's, and 3 5's with a standard one would be (x+2x^3+3x^5)(x+x^2+x^3+x^4+x^5+x^6))