Arranging Marbles in Boxes

Date: 09/16/98 at 19:38:20
From: Chris
Subject: Probability

Six marbles are placed in one of three different boxes. What is the 
probability that each box contains two marbles? What is the formula 
used for this type of problem?

Date: 09/17/98 at 11:21:12
From: Doctor Anthony
Subject: Re: Probability

You can model the situation as follows:

    |****|  |**|      

where the |  | represent boxes and the * are marbles.

In the above diagram we have 4, 0, and 2 marbles in the three boxes, 
respectively.

Now this diagram must begin and end with a  |  but otherwise the two 
remaining |'s and six *'s can be arranged in any order, and each of 
these orders is considered equally likely.

So we have 8 objects to arrange in every possible way, 2 of one kind 
being alike and 6 of a second kind being alike. This can be done in:

     8!
   ------  = C(8,2) = 28 ways.
    2!6!

[ If you haven't done permutations and combinations yet you should 
obtain a textbook on elementary probability and read up on the topic.  
It is essential knowledge for this type of question. There is also some 
information in this archive's FAQ to get you started.

  http://mathforum.org/dr.math/faq/faq.comb.perm.html    ]

So there are 28 possible arrangements of the marbles in the 3 boxes 
and only one of these will have two marbles in each box, so the 
required probability is:

   1/28

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/