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The set "base stat spread" consists of exactly six integers. Each is between 1 and 255 inclusive. The total of all six members of this set must be between 180 and 780 inclusive. Please don't question my curiosity.

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Or "How many different possible"
So ranging from 1/1/1/1/1/1 to 255/255/255/255/255/255, with all numbers in between? Or 180/180/180/180/180/180 to 780/780/780/780/780/780?
I actually have no clue what I'm doing but I plugged 180 and 780 into a combination calculator and got 3.34210013128995E+181 possible combinations.
This had better satisfy Kero.
I will have an answer for you in a few hours. I am currently formulating, and it is complicated because you used a non parallel range. I need to know if you meant BETWEEN 180 and 780 as you said (181-779) or if you meant "The total can be from 180 to 780."

Also, if I am going to put the effort in, I'd like to know the reason. If this is just for curiosity, then I will not share the results/waste my time. So... "Do not question my curiosity" can not apply.
Most likely he just wants to know how many different pokemon, with different stat distributions, GF can make without going below 180 or above 780 BST. He says inclusive so I assume you should count 180 and 780 as well.
It said "inclusive".

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18 trillion, 480 billion, 712 million, 644 thousand 112, give or take.

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Maybe you should provide your working and all, just so that we know you haven't inserted a random number :P
Can't post photos. Basically just adjusted my dice Roller program to use the "exclude" function.
note* H=HPT, A=ATK, D=DEF, S=SAT, d=SDF, s=SPD
-        - I applied that to the following         
H≤255, H≥1
A≤255, A≥1
D≤255, D≥1
S≤255, S≥1
d≤255, d≥1
s≤255, s≥1

180≤(H+A+D+S+d+s)≤780

Then I plotted the potential for a single variable. H returned 154005938700.
       154,005,938,700
*                                   120
______________________
18,480,712,644,112