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Q&A #3398

Teachers' Lounge Discussion: Permutations and counting arrangements

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From: Loyd <loydlin@aol.com>
To: Teacher2Teacher Public Discussion
Date: 2003041409:49:46
Subject: Re: HELP

On 2000101815:38:16, Andrea wrote:
>I am a grade 13 or OAC student in Canada. I am currently taking a
>Finite class the is learning, at the moment, Combinatorics.  I am
>having an extreamly difficult time with the concept of this chapter
>and was wondering if someone would be able to help.  Here is an
>example question that has me stuck...
>
>	
>         Prove:
>
>             1 over (a-1)!n! + 1 over a!(n-1)! = a+n over a!n!
>


Rewrite the problem as:

1              +          1         a+n
-----------          ----------  = ------
(a-1)!n!              a!(n-1)!      a!n!

Multiply 1st term on left by 1 in the form of:

a!(n-1)!
--------
a!(n-1)!

Multiply 2nd term on left by 1 in the form of 

(a-1)!n!
-----------
(a-1)!n!

When you do that you will have


a!(n-1)!           +     (a-1)!n!
-----------            ----------------
a!(n-1)!(a-1)!n!     a!(n-1)!(a-1)!n!

Which reduces to:

a                   +    n
-----              -----------
a!n!                 a!n!

That should do it.  Notice that a!/(a-1)!=a.  That is the key to
solving this problem.  The rest is just solving equations by finding
the LCD.  Pretty tough problem for a 13 year old.

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