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Permutations and Combinations

Date: 10/22/96 at 23:51:21
From: Monique Jain
Subject: Permutations and Combinations

How many different six-digit numbers can be formed using three 5's, 
two 4's, and one 6?

I used 6x5x4x3x2x1 = 720 - is that correct?

Date: 10/23/96 at 5:32:46
From: Doctor Anthony
Subject: Re: Permutations and Combinations

No.  You have 6 numbers altogether to use, and if these were all 
different then the number of different six-digit numbers would be 
6! = 720.  

However, this number is too large because the 5's could be swapped 
amongst themselves without giving rise to a different six-digit 
number, and similarly the 4's could be swapped without producing a 
different six-digit number.  The three 5's could be rearranged in 
3! = 6 ways, and the two 4's in 2! = 2 ways.  So altogether the number 
720 is too large by a factor 6 x 2 = 12.

The correct answer is 720/12 = 60.

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

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