Permutations, Combinations, Arrangements, and Strings
Date: 10/22/2007 at 01:00:20 From: David Subject: permutations vs. multiplication counting principle Many people define permutation as an ordered arrangement of elements when elements are selected without repetition. But suppose you have a 4-question true-false test. Would the possible arrangement of answers, say, TFFT, be a permutation? I think not because the T and F are used with repetition. The number of answer arrangements is found by using the multiplication counting principle (not the permutation formula). This is VERY confusing to students--since order DOES make a difference here. But I do NOT think it is a true permutation. This is akin to a discussion on ordinary combination locks--which, of course do NOT involve combinations. But these are NOT really permutation locks either--because the first and third numbers can repeat. So, I call them "permutation locks--with a twist." The same "confusion" occurs with the usual problems about Heads and Tails. Is an arrangement such as HHHT on four flips a permutation? Even in a context where we would like to distinguish HHHT from, say, HHTH (both have 3 heads and a tail), I would say it's an arrangement, but not a permutation. But, am I too pedantic on this topic?
Date: 10/22/2007 at 13:40:15 From: Doctor Peterson Subject: Re: permutations vs. multiplication counting principle Hi, David. No, you're not too pedantic. It's important to understand that this is not a permutation, because if it were you could use the permutation formula to count them, and you can't. A permutation is, by definition, a selection of k distinct elements from a set, in a specific order. Both distinctness (no repetition) and order are important. Actually, the primary meaning is simply an ordering of (ALL) elements of a set; the permutation formula gives the number of permutations of SUBSETS of a given size. (Combinations are subsets of a given size without regard to order.) See the following page: Wolfram's Mathworld: Permutation http://mathworld.wolfram.com/Permutation.html A result in a true/false test, or coin flipping, or a "combination" on a lock is neither a permutation nor a combination, but a "string", according to this page, which distinguishes four similar concepts: Wolfram's Mathworld: Ball Picking http://mathworld.wolfram.com/BallPicking.html I don't know that I've seen "string" as a general term in this context, but it is certainly the same idea; I think of a string as an ordered list of "letters" from some "alphabet", which fits all your examples. Again, see Wolfram's Mathworld: String http://mathworld.wolfram.com/String.html The following page defines "arrangement" as either a permutation or a combination, the key attribute being distinctness. So your example is not an arrangement in this sense: Wolfram's Mathworld: Arrangement http://mathworld.wolfram.com/Arrangement.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 10/26/2007 at 21:27:55 From: David Subject: Thank you (permutations vs. multiplication counting principle) Dear Doctor Peterson: Thank you so much for your help with respect to my question about permutations / combinations. You have educated me on a new term-- string. I can tell you that I have been teaching stats courses for many years at the community college level and never knew exactly how to refer to this concept. I am delighted to know that I am not "too pedantic" on this topic. David
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