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Permutations


Date: 10/11/97 at 19:19:16
From: Albert Coelho
Subject: Permutations

Dr. Maths,

This question was given to me, as part of my final grade for my 
GCSE'S, and I still cannot understand with only two days to go until 
the deadline,  so this is a matter of emergency!

A number of X's and a number of Y's are written in a row such as

   XX.....XXYY.....Y

Investigate the number of different arrangements of the letters.

Sabrina Coelho


Date: 10/12/97 at 07:06:55
From: Doctor Anthony
Subject: Re: Permutations

For argument's sake suppose you have 9 X's and  6 Y's.

To start with, assume that all the letters are distinguishable, say 
X1, X2, X3, ... , Y1, Y2, Y3, and so on.

If we take the letters one by one at random and place them in a row, 
then we have 15 choices for the first letter, 14 choices for the 
second letter, 13 choices for the third letter and so on. The total 
possible arramngements when all the letters have placed in the row is 
given by:

  15 x 14 x 13 x 12 x ....... x 3 x 2 x 1  =  15! = 1.3076 x 10^12

This would be the answer if all the letters were different. However, 
9 of them are the same of one kind, and 6 are the same of a second 
kind.  This means that the X's could be shuffled amongst themselves 
without actually giving a different sequence, and similarly the Y's 
could be shuffled amongst themselves without producing a different 
sequence. Since there are 9 X's we could arrange them in 9! (= 362880) 
ways, and with 6 Y's we could arrange them in 6! (= 720) ways.

So the first answer, 15!, is too large by factors of 9! and 6!

                                                      15!
It follows that number of different arrangements  = ------  = 5005
                                                     9! 6!


In general if we have n letters, r alike of one kind and n-r alike of 
a second kind, the number of arrangements is given by:

                n!
            ----------
             r! (n-r)!

There is a symbol for this expression, nCr, and you will find it in 
any textbook on permutations anmd combinations.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Permutations and Combinations

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