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Re: combinatorics

• To: mathgroup at smc.vnet.net
• Subject: [mg15189] Re: combinatorics
• From: Daniel Lichtblau <danl>
• Date: Fri, 18 Dec 1998 02:11:05 -0500
• Organization: Wolfram Research, Inc.
• References: <75a4og$sbg@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

mcp wrote:
> 
> problem:a grid with (m+1) row and [m*C(m+1,2)+1] column,to fill the grid
> with m kind of colors, one little grille with one kind color. In the
> grid ,there must be a rectangle of which four corner grilles have same
> color. Why?


There are m+1 rows but only m colors. This means every column will have
at least one color repeated.

There are C(m+1,2) ways to choose a pair of distinct rows, hence
m*C(m+1,2) ways to choose a distinct pair of rows and a color.

Now list, for each column, a pair of rows that have the same color,
along with that color. As per above, there can be at most m*C(m+1,2)
distinct elements in that list. But you have one column more than that,
so there must be a repetition; this gives your rectangle.

I've no idea how this is related to Mathematica. If this is your
homework then I want a part of the grade.


Daniel Lichtblau
Wolfram Research