From: Robin Chapman Newsgroups: sci.math Subject: Re: how many ways to divided 6 students to two classes? Date: Mon, 14 Sep 1998 08:26:07 GMT In article <6th4ou$6s8$1@news.ml.com>, mwang@tech.cicg.ml.com (Michael Wang) wrote: > How:Number of ways > 0+6:1 > 1+5:6 > 2+3:15 > 3+3:20 or 10? Should we have to give special considerations for 3+3? > 4+2:15 If the total number of students is quite large 2*N instead of 6, > 5+1:6 And if we draw a diagram of Number of ways versus the number > 6=1:1 of students in one class, it will look something like a "M" > instead of a reversed "n" and this somehow bothers me. > > OK, The chance that student A and B in the same class for 3+3 > split is 4/20 or 4/10? The number of ways of arranging n objects in k non-empty classes (and assuming the order of the classes is unimportant) is called a Stirling number of the second kind. There are various notations but I'll uses s(n,k) here. When I say order of the classes is not important, I mean that I count say ab|cdef and cdef|ab as the same. There is no general "closed form" for the Stirling numbers, but for each fixed k, there is a closed form in terms of n. Let's consider s(n,2). Let a be one of the n objects, and call the class it falls into class 1 and the other class 2. We now allocate the other n-1 objects into one of the 2 classes. There seem to be 2^{n-1} ways of doing this, but we cannot allocate all the other objects to class 1 lest class 2 be empty. Hence s(n,2) = 2^{n-1} - 1. In particular s(6,2) = 31. There are lots of formulas and identities featuring the Stirling numbers. For many of these see Graham/Knuth/Patashnik, Concrete Mathematics, Addision-Wesley, 1989. Robin Chapman + "They did not have proper Room 811, Laver Building - palms at home in Exeter." University of Exeter, EX4 4QE, UK + rjc@maths.exeter.ac.uk - Peter Carey, http://www.maths.ex.ac.uk/~rjc/rjc.html + Oscar and Lucinda -----== Posted via Deja News, The Leader in Internet Discussion ==----- http://www.dejanews.com/rg_mkgrp.xp Create Your Own Free Member Forum