Dice probabilities

EDIT: finally I fixed all mistakes and I have a deeper understanding on combinatorics.

Of course I did an applet with all the info (I dont know why the applet dont work very fine on the forum... just use the link).

With some more time I will add the sum&reroll mechanic to the mix on the applet (this is very easy to add).

You can see, in general, that the cancel mechanic doesnt change too much the standard probabilities without cancelling, just narrow a bit. The mechanic of cancel compensate a bit the exponential nature of the mechanic without any cancelling.

Thank you for the feedback... I will try to see what is happening. This website have A LOT of applets and I see all ok but maybe Im doing something wrong. When possible I will see how they see in others computers.

EDIT: I think its fixed by now, maybe a problem with too high size on fonts. You cant see it correctly if dont have jre or some java installed. The applet load in two versions: HTML5 that dont charge any LATEX text and is buggy, and java.

P.S.: I updated the others as well.

Im happy it work a bit (I know that for some systems or ipad and similar things it doesnt work because dont use the java environment, it uses instead html5).

I want to write just a last thing that maybe useful to anyone interested on define any type of probability of dice or others games.

The problem for 1's cancellation was really more hard to finalyze that I expected cause my poor understanding on the combinatorics laws.

The key in any problem on combinatorics is to create abstract concepts of independents things, something like a "high level language" to work with "low level" processes (in statistical maths these "low levels" process are named atomic or elemental events).

So if you can define some abstract sets with different composition you can work on this "high level" and the things become easier. In maths this is a norm since Hilbert when he "discover" that you can resolve really hard tasks if you think it from a very high and abstract level where you groups things, and these are the modern maths, i.e., abstract algebra.

In the formulas the binomials (or multinomial) coefficients means permutations with repetitions, i.e., how many shapes or forms I can order a list of elements (in this cases the elements maybe all diferents=normal permutations, or in groups=permutations with repetitions).

And the other big combinatoric concept is variations into a set... the multiplicity of the sets. These are the powers on equations.

So you have two things:

-permutations of sets of things ones with others -> binomial coefficients

-number of variations for each set -> powers

To understand this I have some good examples: variations are combinations from one class of things, a set, something like genetic differences between individuals of the same specie.

And permutations are the amount that different species can divide a territory.

I hope this maybe interesting or useful for someone. Sry if Im too "boring" with these things but I really like a lot maths... I should have been mathematician after all :P

EDIT: last words: the KEY to resolve it was to forget the numbers!!! I imagined the dice instead with numbers just sides of different colours, each colour with a different meaning related to the mechanic (this is why I named the things on the formulas by colours).

From here it becomes more easy...the numbers were confusing me because are not directly related to the meaning on the game or mechanic.

EDIT 2: fixed the representation of probabilities for a fixed value of k and reestructured a bit some info to be more clear.