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Nombre de usuarioGingerAle
Preguntas 2
Respuestas 474


Interesting metaphors.smiley  Generating functions seemed magical to me the first time I saw them demonstrated.  Along with usable numbers popping up like dandelions in May, I actually understood the counting methods used for its derivation, without having to spend hours studying. That was magical for sure.  My mentor, Lancelot Link, who taught the concepts to me wasn’t completely human, and he certainly was from high-up and far-out, but not quite to outer space –though I still do wonder sometimes.indecision


In the case of this OP, while I’m sure he doesn’t (and never will) understand the counting methods for generating functions; he does know what they are. Twice, here and here, he’s referenced this post  https://web2.0calc.com/questions/probability_882, where Nauseated uses a generating function for type 2 Sterling numbers to solve a dice probability problem.


Of course this OP is the one of the forum’s principal (higher functioning) village idiots, the arch idiot, !!!Mr. BB!  (Triple Deranged Mr. BB!). Because he has such an affinity for generating functions, I thought it apropos to use one to solve a combination problem he posted. 


Mr. BB likes to solve math problems using his computer, but his coding skills are as dubious as his math skills and he screws-it-up almost all the time. The “lexicographic” output list, which isn’t lexicographic at all, is his work via his computer.  He looks at this slop output, and though on some level knowing it’s wrong, still believes there are gold nuggets hidden in them-there iterations.  So in the guise of a student, he presents a question on the forum, in the hopes that a high-level mathematician will answer it. ...


...  One did: you. Mr. BB then presents his non-lexicographic, slop output list as an official answer, in the hopes that your advanced skills will magically detect a unique pattern buried in the rubbish that will give it some validity.   Of course, only a god with the power to change mathematical and physical constants could do this; unfortunately, most of them disappeared with the ancient Greeks. Something Mr. BB has lamented cryingabout for years, I’m sure.


 ... And the cycle begins again...





I can offer a guess from where this lexicographic order of BS originated. A mathematics professor usually creates this type of teaching material and he or she will recruit an undergrad (or graduate) student to provide solutions with answers as a paid assignment. The students, and the professors who recruit them, are notorious for not checking and cross checking their solutions for accuracy. (This is why the answers in the “back of the book” are often wrong.)



Obviously, this lexicographic output was generated by a miscoded computerized algorithm that cataloged the correct set along with a massive amount of rubbish. Even if correctly presented, computerized solution algorithms of this type are of limited use beyond verifying a solution.


Along with Rom’s use of binomial counting for each case, there are two other methods for solving this:  A Markov matrix and a generating function. 


Though sometimes difficult to create, the generating function is relatively simple for this application.  For “SSSTTTIIAC” the counts are 3, 3, 2, 1, and 1 for each of the letters. This corresponds to a generating function of (1+x+x^3) (1+x+x^3) (1+x+x^2)(1+x)(1+x)


Expanding this series produces the coefficients.  (Doing this manually takes about 45 minutes. The “Wolf” takes about 10 seconds.)


\(x^{10} + 5 x^9 + 13 x^8 + 24 x^7 + 34 x^6 + 38 x^5 + \mathbf {34 x^4} + 24 x^3 + 13 x^2 + 5 x + 1\\\)


Here, the coefficient of 34 for x4 corresponds to the 34 unique-element subsets of four (4) from a set of 10 elements (S,S,S,T,T,T,I,I,A,C). Note the inclusion of all coefficients for subsets ranging from zero (0) to ten (10).






Note that Rom used engineering notation to depict the numerical values.   Engineering notation is a specialized subset of scientific notation, and its use is common (and preferred) in all branches of physical sciences. 


Reviewing this will help you understand the (different) notations presented by Rom and Melody



Start by familiarizing yourself with the eight prefixes that modify the unit value:

Observe that engineering notation uses base 1000; here it’s contrasted to base 10 used in standard scientific notation.   


tera     T           10004     1012    

giga     G         10003     109     

mega   M         10002     106     

kilo      k          10001     103     

UNIT                10000     100     

milli     m         1000−1   10−3   

micro  μ          1000−2   10−6   

nano    n         1000−3   10−9   

pico     p         1000−4   10−12


Having an instant working knowledge of these prefixes will help you understand these questions.


Another note:

Also, in the Wiki article pay attention to the overview, starting with:

“... Compared to normalized scientific notation, one disadvantage of using SI prefixes and engineering notation is that significant figures are not always readily apparent. ...”   Note in the text, following this extract, that the speed of light is used specifically as an example for significant figures. 


The speed of light constant is used in your question, and though significant figures is not an explicit concern for your presented question or its solution, it is always an implied concern. An understanding of significant figures for constants and measurements will quickly become significant as you pursue your studies in physics.


Other general helps for learning physics:

Learn the Greek alphabet and the context in which the symbols are used (in mathematics and physics). In your question, the Greek letter \(\lambda\) (lambda) is used and its context means Wave Length. The Latin letter (c) is used to denote the constant for the speed of light in a vacuum.  


Remember this constant (c) is speed of light in a vacuum; when light passes through other media, it slows down, and further, it slows down in proportion to its frequency (wave length). This is why you can see “rainbow” colors when composite (white) light passes through the atmosphere or a prism.     


Another thing to remember is “light” in this context refers to the full electromagnetic radiation spectrum, not just the small portion that is visible.