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  1. #21
    no clinkz 'til brooklyn Nocapszy's Avatar
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    Quote Originally Posted by ygolo View Post
    If I'm reading you correctly, it is a matter of combinatorics. I may have interpreted you incorrectly but I agree with Urchin.

    Think of this way, you have 3 options for a shirt (first slot), 3 options for pants (second slot), and three options for a tie (third slot).
    This analogy doesn't allow you to use all three shirts and no pants or no ties. I mean... you could, but then why bother with clothes?

    You have three independent choices and three things to choose from for each choice. So yes, it is 3^3.
    Yeah that's what I was thinking.
    But the order-independent version can be thought to be equivalent to the following:

    You have a giant(infinite) vat of ping-pong balls numbered 1 to 3, and choose 3 ping pong balls from there. You subsequently forgot which order you chose them. How many distinguishable possibilities are there?

    It is 10, like Urchin mentioned,
    What if you got two 3 balls? Or 3 twos? Then it would be 27 again.
    but how do you do it more generally? With choosing r balls from n types of balls? That is an interesting question, imo.

    The short answer is it is C(n+r-1,r), where C is the choose function.

    I tried typing the "star" and "bar" explanation but I didn't like the formating.

    So, here is a link.

    Nocapszy, if you like these types of questions you'll like Discrete Math (which is the mathematics behind computer science).
    I actually have been toying with things like this (binary mostly though) and you're right. I do like that kind of math, if any. It's a really layered math - fun for Ne and/or Ti.
    The following source has explanations of the basics.
    Discrete Math Project - Permutations with Repetition

    Also note, that these basics come in four flavors:
    1)permutations without replacement
    2)combinations without replacement
    3)permutations with replacement
    4)combinations with replacement

    You'll find that transforming your counting problem into one of these is the most likely source of answers. That along with generating functions will fulfill most people's combinatorics type questions.
    Thank you.
    we fukin won boys

  2. #22
    Senior Member Urchin's Avatar
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    Quote Originally Posted by Nocapszy View Post
    This analogy doesn't allow you to use all three shirts and no pants or no ties. I mean... you could, but then why bother with clothes?
    This is the same logic you used, but possibly more obfuscated. What he's saying is you worked the problem thus: You have "spot one," "spot two," and "spot three," then there are three choices for each spot. 3^3.

    However, this will cause the set {3, 3, 1} to be counted as different from the set {1, 3, 3} and {3, 1, 3}. These are permutations.

    You're looking, I think, for a combination. This would cause the three sets I mentioned to be counted as the same thing because they all have two 3s and one 1. In this case you have:

    333, 332, 331, 322, 321, 311, 222, 221, 211, 111

    Which is 10 combinations.
    "Having is not such a pleasing thing as wanting. It is not logical, but it is often true." --Spock

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  3. #23
    Enigma Nadir's Avatar
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    Even simpler 27-set analogy (though I'm not sure if it works...):

    You have three sets of three coins. The coins are made of gold, silver, and copper, and each coin represents a value of 100, 10, or 1. Let's say you have a purse which can take 3 coins at a time. There are 27 different possibilities for you involving depositing your money.

    For the 10-set example, consider all coins in a single set to have equal worth.
    Then you have the following possibilities:

    1. Gold, Gold, Gold
    2. Gold, Gold, Silver
    3. Gold, Silver, Silver
    4. Silver, Silver, Silver
    5. Silver, Silver, Copper
    6. Silver, Copper, Copper
    7. Copper, Copper, Copper
    8. Copper, Copper, Gold
    9. Copper, Gold, Gold
    10. Gold, Silver, Copper
    Not really.

  4. #24

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    The last one is a good analogy.

    Counting permutations would be counting the distinguishable ways of putting the coins in the purse.

    But counting combinations would be counting the distinguishable states when looking in the purse after the coins got all jumbled up.

    You'd have no way of knowing which way the coins went in if you only saw them in the purse afterwards.

    I hope the difference between combinations and permutations makes sense now.

    Accept the past. Live for the present. Look forward to the future.
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    "As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
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  5. #25

    Default Group Theory, Symmetry, Super-Symmetry, and M-Theory

    Since the combinatorics related discussion on this thread has waned...

    I was wondering what you guys knew about symmetry groups, their relation to symmetry in general, supersymmetry, and M-Theory.

    I believe understanding lie algebra and gauge theories are an important part of understanding these things.

    There is a lot of hefty math in there, and String Theory appeals to many people who aren't mathematically inclined. So if there is a way to transform the math to some pictures or some other form of intuitive understanding, I think it would go a long way to actually understanding string theory (as opposed to just being aware of it).

    Also, I like knowing the "skeletal rigorous framework" for a mathematical ideas. By this I mean, the minimum set of definitions and theorems needed to understand a mathematical idea. But I haven't figured out what it is for M-Theory yet.

    Anyone care to collaborate on this thread to create an informal, multi-type friendly, math-flavored exposition?

    Accept the past. Live for the present. Look forward to the future.
    Robot Fusion
    "As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
    "[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

  6. #26
    no clinkz 'til brooklyn Nocapszy's Avatar
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    Quote Originally Posted by Urchin View Post
    This is the same logic you used, but possibly more obfuscated. What he's saying is you worked the problem thus: You have "spot one," "spot two," and "spot three," then there are three choices for each spot. 3^3.
    It's not quite the same. The permutations you bring up are considered different possibilities (I was figuring it out for a game I play, but then I just liked the math part and wanted to show you guys)

    However, this will cause the set {3, 3, 1} to be counted as different from the set {1, 3, 3} and {3, 1, 3}. These are permutations.
    Yeah. Each permutation counts as a different possibility in the game. I see what you're saying though and also how it would be confused with a match for ygolos points.

    You're looking, I think, for a combination. This would cause the three sets I mentioned to be counted as the same thing because they all have two 3s and one 1. In this case you have:
    No. You had it right a second ago.


    Edit: The game actually does go by combination, and not by the permutations. The way my friend explained it though, it sounded the other way. One way or another, my question here was posed searching for all combinations and incarnations of that combination as separate.
    we fukin won boys

  7. #27

    Default MBTI Types, Hypercubes, and Hamiltonian Paths

    Consider the following 4-dimensional hypercube with MBTI types on each node:

    MBTIc Math thread-mbti_hypercube-jpg

    A Hamiltonian Path is (informally) a way to visit each node/vertex in a graph without picking up your pencil.

    Can you find all Hamiltonian Paths on the MBTI hyper-cube?

    One such path was used in my post on MBTI superlatives.

    How many different such Paths are there?
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    Accept the past. Live for the present. Look forward to the future.
    Robot Fusion
    "As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
    "[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

  8. #28
    Senior Member nemo's Avatar
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    Quote Originally Posted by ygolo View Post

    nemo, do you have any clear picture/understanding of Stone-Weierstrass or the Baire Category theorems?
    Not really. I haven't really taken a proper analysis course yet -- the one I took this semester was an undergraduate intro to analysis/set theory combined course, and a lot of the philosophy and historical context was integrated into the material. But I had a blast with it.

    Next fall is when I start the joint undergraduate/graduate real & complex analysis sequence. I'll be starting abstract algebra and topology at the same time. I'd take it now, but my school is sufficiently small that those upper-level sequences are only offered once a year.

    Ironically, the one place where I have encountered the Stone-Weierstrauss Theorem was in a numerical analysis course. But we didn't get into the theoretical aspects enough to really give me any kind of understanding of it.

  9. #29
    Senior Member Gabe's Avatar
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    What in the world is a hyper cube?
    Jumping in late here, but here we go: I only know a bit of the highly theoretical stuff. I like the gradient devinition, and the idea of potential functions and conservative fields (whichever theorum that stuff is part of).
    Mostly in terms of how it applies to physics and the real world.
    Oh, and I still have trouble with imaginary numbers. What is the deal with those?

  10. #30
    にゃん runvardh's Avatar
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    square n^2
    cube n^3
    hypercube n^4
    Dreams are best served manifest and tangible.

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    I accept no responsibility, what so ever, for the fact that I exist; I do, however, accept full responsibility for what I do while I exist.

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