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#1 2007-06-01 03:45:08

Identity
Member
Registered: 2007-04-18
Posts: 934

Arrangements

Find all possible arrangements of the word PIGEON so that the vowels are in alphabetical order.

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#2 2007-06-01 04:07:26

Daniel123
Member
Registered: 2007-05-23
Posts: 663

Re: Arrangements

EIOPGN     PEIOGN     PGEION     PGNEIO     EPIOGN     EIPOGN     
EIOPNG     PEIONG     PNEIOG     PNGEIO     EPIONG     EIPONG
EIOGPN     GEIOPN     GPEION     GPNEIO     EGIOPN     EIGOPN
EIOGNP     GEIONP     GNEIOP     GNPEIO     EGIONP     EIGONP
EIONPG     NEIOPG     NPEIOG     NPGEIO     ENIOPG     EINOPG
EIONGP     NEIOGP     NGEIOP     NGPEIO     ENIOGP     EINOGP

EPGION     EPGNIO     EIPGON     EIPGNO     PEIGNO     PGEINO
EPNIOG     EPNGIO     EIPNOG     EIPNGO     PEINGO     PNEIGO
EGPION     EGPNIO     EIGPON     EIGPNO     GEIPNO     GPEINO
EGNIOP     EGNPIO     EIGNOP     EIGNPO     GEINPO     GNEIPO
ENPIOG     ENPGIO     EINPOG     EINPGO     NEIPGO     NPEIGO
ENGIOP     ENGPIO     EINGOP     EINGPO     NEIGPO     NGEPIO

Thats all I can be bothered to do. There is more....

Edit: Assuming Mathsyperson is right, then there is another 48

Ok I might as well add the rest (thaks mathsy):

EPIGON     EPIGNO     EPGINO     PEIGNO     PEGION     PEGINO       
EPINOG     EPINGO     EPNIGO     PEINGO     PENIOG     PENIGO
EGIPON     EGIPNO     EGPINO     GEIPNO     GEPION     GEPINO
EGINOP     EGINPO     EGNIPO     GEINPO     GENIOP     GENIPO
ENIPOG     ENIPGO     ENPIGO     NEIPGO     NEPIOG     NEPIGO
ENIGOP     ENIGPO     ENGIPO     NEIGPO     NEGIOP     NEGIPO

PEGNIO     PGENIO
PENGIO     PNEGIO
GEPNIO     GPENIO
GENPIO     GNEPIO
NEPGIO     NPEGIO
NEGPIO     NGEPIO

That's all 120.

Last edited by Daniel123 (2007-06-01 04:31:34)

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#3 2007-06-01 04:08:33

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Arrangements

...Except that there's an I in there as well, that also needs to be ordered correctly.
So the actual answer is 6!/6 = 120.

Edit: Daniel's well on the way to showing the list exhaustively. By the way he's grouped them, it's obvious that there are 6 combinations for every way that you can specifically place the E, I and O.

eg. there are 6 of the form EIO###, 6 of the form EI#O##, etc.

In addition to what he's already got, the additional forms are:

E#I#O#
E#I##O
E##I#O
#EI##O
#E#IO#
#E#I#O
#E##IO
##E#IO

Daniel had 12 forms and I've added another 8, so that means that there are (12+8)x6 = 120 combinations again.

Last edited by mathsyperson (2007-06-01 04:16:56)


Why did the vector cross the road?
It wanted to be normal.

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#4 2007-06-01 04:26:36

Identity
Member
Registered: 2007-04-18
Posts: 934

Re: Arrangements

Right, thanks everyone

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#5 2007-06-02 06:59:01

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Arrangements

Here is the general case. Suppose you have n letters (all distinct) and you want to find the number of permutations in which r of them are ordered in a specific way.

Well, you first choose r spots from the n places in which to put your r letters in their specific order. This can obviously be done in [sup]n[/sup]C[sub]r[/sub] ways. For each of these [sup]n[/sup]C[sub]r[/sub] placements, the other nr letters can be placed in the other spots in (nr)! ways. Hence the total number of ways of arranging n letters with r of them in a specific order is

Last edited by JaneFairfax (2007-06-02 07:10:48)

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