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Combinations of X's and Y's

Date: 10/27/1999 at 09:58:42
From: John Dodgson
Subject: GCSE coursework (arrangements and formula)

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 and find a formula for it. 

I tried writing out the different arrangements, but I'm not sure of 
the arrangements or the formula for the number of the arrangements. 
Here are some of the formulas I have come up with:

     C = n^2
     C = x!/y!

     C = n! / [(number of X's)! * (number of Y's)!]

Please can you help me?

Date: 10/27/1999 at 11:42:53
From: Doctor Anthony
Subject: Re: GCSE coursework (arrangements and formula)

Let's begin with a general note on permutations and combinations from 
our archives:

  Permutations and Combinations: Deriving Formulae    

So if you have m X's and n Y's, in how many DIFFERENT ways could these 
m+n letters be arranged?

In the case of m X's and n Y's, if all the X's were different and all 
the Y's were different there would be m+n different letters to 
permute, and this could be done in (m+n)! ways. 

However, since the X's are all the same, they could be permuted 
amongst themselves in m! ways and the answer (m+n)! is too large by a 
factor of m!. 

Similarly, the Y's are all the same and could be permuted amongst 
themselves in n! ways without giving rise to a new arrangement, and 
(m+n)! is too large by a further factor of n!. So the actual number of 
different arrangements is:

       m! n!

Another example: in how many different ways can the letters of the 
word STEEPLES be arranged?

There 8 letters. 3 E's, 2 S's and 3 other letters.

Number of arrangements =  ----- = 3360
                          3! 2!

- Doctor Anthony, The Math Forum   
Associated Topics:
High School Permutations and Combinations

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