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Combinations of X's and Y's

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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!
and

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

Thanks
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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
http://mathforum.org/dr.math/problems/lim6.18.98.html

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)!
--------
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.

8!
Number of arrangements =  ----- = 3360
3! 2!

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Permutations and Combinations

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