The full question "Given variables that can take on certain values, and the sum of the values has to equal the determined total, how would you find the number of combinations that equal the total value and the number of permutations of the combinations?"
Imagine four particles within a box. The total energy of the particles can not exceed 5 units, for now it'll be MeV (Mega Electron Volts). Each particle can take on a value of 0≤X≤5 (Energy Level 0-5). There are six possible states (Ps) that these particles can be arranged in.
Ps One (P(1)): Particle One (Part1) through Part3 could be at energy level 0, while Part4 would be at energy level 5.
P(2): Part1-3 at energy level 1. Part4 at energy level 2.
P(3): Part1-2 at energy level 0. Part3 at energy level 1 and Part4 at energy level 4.
P(4): Part1-2 at energy level 0. Part3 at energy level 2 and Part4 at energy level 3.
P(5): Part1 at energy level 0 and Part2-3 at energy level 1. Part4 at energy level 3.
P(6): Part 1 at energy level 0 and Part2 at energy level 1. Part3-4 at energy level 2.
Now we come order. Each Ps (Possible State) has multiple possible arrangements when it comes to the order of the particles. These arrangements will from here forth be called microstates, Ms.
P(1)=4 Ms; P(2)=4 Ms; P(3)=12 Ms; P(4)=12 Ms; P(5)=12 Ms; P(6)=12 Ms.
This brings total number of mircostates to 56.
So, given the number of particles, x; the number of energy levels, y; and the the total energy level, z MeV; how would one determine the number of possible states and mircostates?
Also, the occurrence of particles within energy level 0 for P(1) is equal to the number of particles times the number of microstates, (3*4=12). This would be 12 occurrences within energy level 0 for P(1). Can you determined the total occurrences for each energy level for each possible state given x,y,z?
Also, would there be anyway to do this on a ti-84, specifically a ti-84 CE Plus? I'm open to writing custom programs or using preset functions.