       RE: Combinations

• To: mathgroup at smc.vnet.net
• Subject: [mg30776] RE: [mg30769] Combinations
• From: "David Park" <djmp at earthlink.net>
• Date: Wed, 19 Sep 2001 00:16:16 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Dana,

The number of permutations of n objects is just n! (n factorial) which you
can enter in just that way in Mathematica. For a set with 4 elements:

4!
24

Which checks with

Permutations[{1, 2, 3, 4}]
Length[%]
{{1, 2, 3, 4}, {1, 2, 4, 3}, {1, 3, 2, 4}, {1, 3, 4, 2}, {1, 4, 2, 3}, {1,
4,
3, 2}, {2, 1, 3, 4}, {2, 1, 4, 3}, {2, 3, 1, 4}, {2, 3, 4, 1}, {2, 4, 1,
3}, {2, 4, 3, 1}, {3, 1, 2, 4}, {3, 1, 4, 2}, {3, 2, 1, 4}, {3, 2, 4,
1}, {3, 4, 1, 2}, {3, 4, 2, 1}, {4, 1, 2, 3}, {4, 1, 3, 2}, {4, 2, 1,
3}, {4, 2, 3, 1}, {4, 3, 1, 2}, {4, 3, 2, 1}}
24

The number of k element subsets picked from n items is just Binomial[n, k].
For example:

Binomial[4, 2]
6

Needs["DiscreteMath`Combinatorica`"]

KSubsets[{1, 2, 3, 4}, 2]
Length[%]
{{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}
6

How precisely did I find this out? By looking up Permutations in my
"Mathematics Handbook for Science and Engineering" and then looking up
Binomial and Factorial in Help. Finding KSubsets is not that easy because
the Combinatorica Help is not very good or convenient.

There was a book "Implementing Discrete Mathematics: Combinatorics and Graph
Theory with Mathematica" by Steven Skiena the author of Combinatorica.
Unfortunately, it has gone out of print. It would be a great thing if WRI
could somehow arrange to bring it back into print, and also improve the Help
documentation for Combinatorica because it is a great package.

New users of Mathematica are often disappointed that the precise command or
function they want is not there, or not there under the name they might
expect. But you will quite often have to hit the math books to find how to
calculate something and you will also quite often have to write your own
small routines. If Mathematica tried to have direct routines for everything
it would end up with millions of commands and still fall short.

David Park

> From: Dana [mailto:ng_only at hotmail.com]
To: mathgroup at smc.vnet.net
>
> Hello.  I have Mathematica 4.1
>
> In a Program like Excel, or a hand-held calculator, one can return the
> number of combinations and permutations.
>
> However, I can not find an equivalent function in Mathematica.
> For example, Permutations[ ] returns a long list of all the "Actual"
> permutations.
> I am looking for just the final number.
> If there is one, could you include 'how' you found it.  I have looked
> everywhere.
> I know DiscreteMath`Combinatorica` has some stuff in it, but the
> Help system
> appears not to explain many of them.
>
> I can write a custom function, but I am curious to find out if
> this function
> is built in to Mathematica.
>
> I hope the answer is not to take the Length[ ]  of a rather long list.
>
> TIA.  Dana
>
> (I posted this question a month ago, but it never showed up in the
> newsgroup, or in the archives.
> I hope I am not doing something wrong.)
>

```

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