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Permutations of 1234567890

Date: 09/23/2001 at 06:09:19
From: Leeanna blyton
Subject: Combinations

I'm trying to find a pattern in combinations and how many combinations 
there are in 1234567890.
I have found that 

   7 = 1   46 = 2   123 = 6   4567 = 24 

I am now trying to find out how many combinations there are in 
5-, 6-, 7-, 8-, 9-, and 10-digit numbers.


Date: 09/23/2001 at 07:19:34
From: Doctor Mitteldorf
Subject: Re: Combinations

Dear Leeanna,

You're doing it just right. Doing examples with small numbers of 
digits will help you to see a pattern, and then you'll know how to 
calculate the combinations for any number of digits. (Actually, these 
are "permutations," not "combinations." For a review of Permutations 
and Combinations, see the Dr. Math FAQ:     )

The hard part about it is that the numbers get big so fast. You can 
list the permutations for 4 without too much trouble, but there are 
enough of them for 5 that you might have trouble listing them all 
without making mistakes. So the key is to come up with some kind of 
system so you're sure you have them all.

Here's a suggestion for a system. Say you want to find the 
permutations of the digits 12345. Let's keep the 5 anchored at the end 
and list all the permutations that end in 5. Well, what you're left 
with is the digits 1234, and you can put them in any order. So you 
know how many there will be - you just did this problem, and found 24.  

Now let's count how many there are that end in 4. The other digits are 
1235, and they can be in any order. But this is really the same as the 
problem you just did: there are 24 ways to order the numbers 1235, so 
there are 24 permutations of the digits 12345 that all end in 4.

Now you're starting to see a pattern, I think. Follow the reasoning 
and you can count how many permutations there are going to be 
altogether without having to list them all.

Once you've done this, you know the answer for 5 digits. Let's move on 
to 6. Can you apply the same kind of reasoning? How many permutations 
are there that end in 6? Well, you just calculated that by your astute 
and careful reasoning in the problem above. Then you list each of the 
other digits that the permutations could end in, and you have an idea 
how to get the total for 6 digits. 

Getting really abstract, you might formulate a rule: If I know how 
many permutations there are for N digits, then I can figure how many 
there are for N+1 digits. Describe the rule for how you do that. Make 
a chart, like the one you started, for the number of permutations of 
1,2,3,4,5 6 and 7 digits.

(Now I don't want you to peek. Finish the whole process that I've 
described above, and understand the patterns that you see. Only after 
you understand just what you've done and why it comes out the way it 
does, go to your dictionary and look up the word "factorial.")

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

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