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### 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
```
Associated Topics:
College Definitions
High School Definitions
High School Permutations and Combinations

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