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 > > task: > > 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. > >>> > >>> > >>> > >>> > > > > > > > >