Re: Combinatorica questions!!!

• To: mathgroup at smc.vnet.net
• Subject: [mg20656] Re: [mg20499] Combinatorica questions!!!
• From: Rob Pratt <rpratt at email.unc.edu>
• Date: Sun, 7 Nov 1999 02:09:58 -0500
• Sender: owner-wri-mathgroup at wolfram.com

```Equivalently,

Multinomial[7,7,7]

gives 399072960.

The generating function is (x + y + z)^21.  The coefficient of
x^7 y^7 z^7 is 399072960.

Rob Pratt
Department of Operations Research
The University of North Carolina at Chapel Hill

rpratt at email.unc.edu

http://www.unc.edu/~rpratt/

On Thu, 4 Nov 1999, Andrzej Kozlowski wrote:

> It seems clear that Rob Pratt's method applies to this also: the rook has
to
> make 21 moves, 7 in the x direction, 7 in the y direction and 7 in the z
> direction. So we need to choose seven out of the 21 moves to be in the x,
> direction and for each of such choices we need to choose 7 out of the
> remaining 14 as moves in the y direction. The rest will automatically be
> moves in the z direction. So the answer is:
>
> In[6]:=
> Binomial[21, 7]*Binomial[14, 7]
> Out[6]=
> 399072960
>
>
>
> --
> Andrzej Kozlowski
> Toyama International University
> JAPAN
> http://sigma.tuins.ac.jp
>
>
> > From: kewjoi at hixnet.co.za (Kew Joinery)
To: mathgroup at smc.vnet.net
> > Reply-To: kewjoi at hixnet.co.za
> > Date: Thu, 04 Nov 1999 10:30:35 +0200
> > To: "mathgroup at smc.vnet.net" <mathgroup at smc.vnet.net>
> > Cc: Rob Pratt <rpratt at email.unc.edu>, Andrzej Kozlowski <andrzej at tuins.ac.jp>
> > Subject: [mg20656] Re: [mg20499] Combinatorica questions!!!
> >
> > Hello,
> > The case has been solved perfectly well. Can you do so for slightly

> > different
> > Same conditions, but for 3 dimensional 8x8x8 chessboard (cube). Imagine that
> >
> > the rook can move not only on the surface but inside the cube too.
> > To make it clear I will denote the start position of the rook {0,0,0}. The
> > target
> > is final position {7,7,7} which is the farthest opposite point. The rook can
> > move
> > as usual (not diagonally), the only constraint is you can move the rook
in
> > increasing order of each coordinate.
> > Example for allowable move:
> > Say from {0,0,0} to {0,6,0} but not {0,6,6},
> > Say from {4,3,5} to {4,7,5} but not {4,1,5}.
> > In other words: the change of only one coordinate at a time equals one
move of
> > the rook, and the change could be in increasing order of each coordinate!
> > *** The task is how many different ways (walks) does a castle have to reach
> > from
> > position {0,0,0} to position {7,7,7}?
> > (**Is there a general formula or generating function for higher dimension?  )
> > (Note: some people could find the question not relevant to the group, but this
> > is
> > pure mathematics and this is just the beginning of the difficult questions and
> > answers normally are available only to research people as usual, so everyone
> > could learn something positive).
> >
> > Thank you.
> > Eugene
> >
> > Rob Pratt wrote:
> >
> >> An approach to problem 1 that is simpler than those already given is to
> >> recognize that each path consists of a sequence of 14 moves, 7 of them
to
> >> the RIGHT one space and 7 of them UP one space.  Hence a path is uniquely
> >> determined by specifying which 7 of the 14 moves are RIGHT (the rest are
> >> UP).  We are choosing 7 objects from among 14 positions, so the answer is
> >>
> >> Binomial[14,7]=3432
> >>
> >> Rob Pratt
> >> Department of Operations Research
> >> The University of North Carolina at Chapel Hill
> >>
> >> rpratt at email.unc.edu
> >>
> >> http://www.unc.edu/~rpratt/
> >>
> >> On Wed, 27 Oct 1999, Keren Edwards wrote:
> >>
> >>> Hi all!!
> >>>
> >>> 2 different questions:
> >>>
> >>> 1.    how many ways does a castle have to reach from the bottom left
side
> >>> corner
> >>> of a chess board to the upper right corner of the board if he can
> >>> move right
> >>> and up only?
> >>>
> >>>
> >>>
> >>> 2.     you have 8 red identical balls, 9 purple identical balls and 7 white
> >>> identical ones.
> >>> a.  How many ways can you choose 10 balls with no matter to the
> >>> order of the balls?
> >>> b.  How many ways can you choose 10 balls with no matter to the
> >>> order of the balls, if each color must
> >>> be chosen once at least?
> >>>
> >>>
> >>>
> >>> Many thanx.
> >>>
> >>>
> >>>
> >>>
> >
> >
> >
>
>

```

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