@Kitonum The OP's underlying question is whether the results were achieved by a random selection. This is a claim that can be tested by standard hypothesis testing. The p-value of the test is the probability of achieving any team split at least as extreme as the observed split. If the split was numerically the same but to differently colored teams, that would be a split as extreme as the observed one. The observed case is numerically the most extreme possible, so we don't need to consider any even-more-extreme cases. The computation I gave gives the p-value. It equals 6/1001.

@Carl Love However, my computatIon will not help you solve your problem. Rather, its purpose is to show that the symbolic solution to your problem would be much too large to be comprehended by the human mind, so maybe you should seek a numeric solution instead. The actual symbolic solution to your problem would be many, many times larger than the solution to the toy problem that I created from your problem.

@asma khan Use a seq command inside the angle brackets, like this:

<seq(x[i], i= 1..N)>

@mmcdara You show a potentially useful way of duplicating this variable-in-subscript-is-constant partial derivative notation that I was previously unfamiliar with but which is well established in thermodynamics. However, like Tom, you've missed that this is a problem requiring implicit differentiation.

[The paragraph above only applies to an earlier version of your Reply above which did not give the implicit derivatives.]

My question in the Reply immediately above was rhetorical, and the OP indicated (in a misplaced Reply to the main Question) that they understood how to modify my Answer to get the first implicit partial derivatives instead of the mixed second implicit partial derivatives that I originally coded. Unfortunately, the MaplePrimes timestamps are not precise enough to reveal this logical order. The StackExchange software is so much better than MaplePrimes for a Q&A site--like a hundred times better.

@dharr It is very likely that this "variation of iteration" algorithm could be implemented in Maple, so it's well within the scope of MaplePrimes. The best way to proceed would be to post a link to a reference to the algorithm.

@reza gugheri I will post that, but it'll be several hours before I get to my computer to do it. I'm posting from my phone now.

@reza gugheri You have a system of 4 polynomials that each are degree 3 in the 4 decision variables and have coefficients that are lengthy rational functions. I made a much-simplified version of the system by replacing each rational function coefficient with a small random integer. Maple 2019 can solve this new system in under a minute, but each solution (other than the trivial solution) is about 14 million bytes long, and some coefficients in the solutions are integers with about 9000 digits.

@mehran rajabi Since partial derivatives always assume that the other variables are constant, what's the point of stating "when V is constant" and "when T is constant"?

@mehran rajabi And you can't figure out how to modify my Answer to get those? 

@tomleslie You've taken the derivative of both sides of the equation treating P as a constant or independent variable. That's not the same thing as (d^2 P) / (dT dV), etc. If you want to do it with diff rather than implicitdiff, you need to first substitute P(T,V) for P.

Since the first two of your three derivatives are necessarily equal (in symbolic computation), perhaps I don't understand your notation. Doesn't your Diff(P,T)[V] just mean Diff(P, [T,V])?

Thank you both. Control-Shift-R works as expected.

@reza gugheri Acer only means that he put a notation at the top of your Question as it appears on this site indicating that you are using Maple 13. No-one has yet reached a definite answer as to whether your problem can be solved by any version of Maple. I'm working on it, and my preliminary answer is that it can be solved (in terms of very complicated RootOf expressions), but the answers may be too long to be of much value.

@reza gugheri You've posted a worksheet and asked for an "answer" but you haven't asked a question.

@gaurav_rs Your input expression F1 has (if I counted correctly) 33 digits of precision in its coefficients. If you have Digits set to 10 when you input the expression, of course you'll get inaccurate results. My fast symbolic method can get you the full 33 digits of accuracy, which, of course is the most that you can expect given the coefficients. This accuracy can be confirmed by numeric integration, which is very robust for accuracy.

restart;
Digits:= 35:
F1:= #Lengthy input expression omitted.
#Fully symbolic integration (fastest way):
CodeTools:-Usage(
   int(fnormal(F1), [theta= Pi/4..7*Pi/4, r= 0..1])
);
memory used=57.39MiB, alloc change=-4.00MiB, cpu time=734.00ms, real time=736.00ms, gc time=125.00ms
                                                   12
           -1.1160010071917964702041524480427751 10  
#Inner integral symbolic, outer integral numeric:
CodeTools:-Usage(
   int(int(fnormal(F1), theta= Pi/4..7*Pi/4), r= 0..1, numeric)
);
memory used=169.19MiB, alloc change=140.01MiB, cpu time=1.59s, real time=1.81s, gc time=125.00ms
                                                   12
           -1.1160010071917964702041524480427751 10  
#Same thing with order of integration switched:
CodeTools:-Usage(
   int(int(fnormal(F1), r= 0..1), theta= Pi/4..7*Pi/4, numeric)
);
memory used=0.73GiB, alloc change=51.13MiB, cpu time=5.09s, real time=4.83s, gc time=843.75ms
                                                   12
           -1.1160010071917964702041524480427751 10  
#Fully numeric double integration:
Digits:= 15:
CodeTools:-Usage(
   int(fnormal(F1, 35), [theta= Pi/4..7*Pi/4, r= 0..1], numeric)
);
memory used=277.90MiB, alloc change=168.00MiB, cpu time=5.03s, real time=5.01s, gc time=234.38ms
                                         12
                     -1.11600100719180 10