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Maximum possible 10 character string combinations from 36 unique characters

Posted on 2016-09-30
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I am trying to determine what is the maximum number of 10 character word string combinations that can be created using 36 unique characters. For example, let's say that you use the English alphabet, all lowercase letters, a-z, and all the Arabic numerals, 0-9. You would end up with 36 unique characters to work with:

abcdefghijklmnopqrstuvwxyz0123456789

With these 36 characters, how many unique 10 character strings can be created?  What would be the math formula that would be used to determine the answer to this question?

You could start at:

aaaaaaaaaa

then iterate to.... aaaaaaaaa1, then aaaaaaaaa2, etc...

eventually, the last possible 10 character string would be... 9999999999

If you tallied up all the possible combinations, how many would there be?

Would it 10 to the 36th power?
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Question by:Fontographer
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d-glitch earned 500 total points
ID: 41823983
36^10

(36 choices for the first character) x (36 choices for the second) ....
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by:Shaun Kline
ID: 41823987
No.

Consider a simpler concept: just two characters.

If each space in the word can have 36 characters, the initial character in the second space in the word can have 36 unique combinations. When the second space changes to a new unique character, that character can also have 36 unique combinations. Following this pattern, you would find you would have 36 x 36 combinations.

For 10 spaces in the word, you would have 36 x 36 x..., or 36 to the 10th power.
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by:d-glitch
ID: 41823988
= 3.656 x 1015
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by:pony10us
ID: 41824033
I believe the answer would be 254,186,856 possible combinations using 10 out of a possible 36 characters.

If you are speaking of permutations then the result is much different:   922,393,263,052,800

This is because in a combination the result of 1234567890 would be the same as 2345678901 but in a permutation they would be different.  Combinations don't care about the order.
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by:pony10us
ID: 41824054
You can find a good explanation of permutations and combinations, including formulas and samples, at Math Planet:

http://www.mathplanet.com/education/algebra-2/discrete-mathematics-and-probability/permutations-and-combinations
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Author Comment

by:Fontographer
ID: 41824062
Clarification:

This question has to do with filenames on a Windows computer. If you want to know how many unique 10 character filenames (excluding any file extension), what would be the answer?  Yes, I know all about the limitations of how many files a given folder can contain, but ignoring any limitation of the Windows operating system, what would be the largest number of filenames possible using only alphanumeric characters?
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by:d-glitch
ID: 41824068
I believe the answer would be 254,186,856 possible combinations using 10 out of a possible 36 characters.

That seems very low.  What is your formula.

And you can repeat characters, so permutations and combinations are not relevant.

So what's wrong with    1036 = 3.656 x 1015
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by:pony10us
ID: 41824097
combination formula:  

C(n,r)
C(36,10)
36! / (10! (36 - 10)!)
2.54186857 E+8
254,186,856
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by:pony10us
ID: 41824106
I have verified that amount through three different online calculators as well.
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by:pony10us
ID: 41824113
Okay,  I just tried one more online calculator and did come up with a different total but not by much:

Types to choose from? 36
Number Chosen? 10
Is Order important? No
Is Repetition allowed? Yes

Short Answer: 3.190187286e+9
Full Answer: 3190187286

Site used:  https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html
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by:d-glitch
ID: 41824132
Order is important . . .
And repetition is allowed.

Full Answer:  3.6561584400e+15  as I said.
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by:pony10us
ID: 41824146
Okay, so by making the order important you have brought it back to being a permutation instead of a combination. That's what I stated before.  Granted my calculations may be off but 3 out of 4 online calculators agree with my previous calculation for permutation:

P(n,r)
P(36,10)
36! / (36 - 10)!
9.22393263 E+14
922,393,263,052,800  

The site I referenced above does come back with the same result as you provided.
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by:d-glitch
ID: 41824158
Permutations and combinations are not relevant when repetition is allowed.

The website will give you the correct answer if you give it the correct input:
     Balls / 36 / 10 / Yes / Yes

Even the formula changes:  nr ==>  1036 = 3.656 x 1015  as I said two hours ago . . .
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by:d-glitch
ID: 41824173
Which websites give different results?
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by:pony10us
ID: 41824183
No - order is the difference between combination and permutation.  

Don't care about order = combination
Care about order = permutation

There are two different types of permutations

Repetition allowed
Repetition not allowed

Now having gone through all that - haven't done this in a long time - I see where this question is actually looking for a permutation with repetition allowed (otherwise aaaa... wouldn't be possible) so yes your total is correct.  My total for permutation was incorrect because it was not allowing repetition.

Thank you for this, it brings back some memories.
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by:pony10us
ID: 41824196
To answer about the other sites:

http://www.calculatorsoup.com/calculators/discretemathematics/permutations.php
http://stattrek.com/online-calculator/combinations-permutations.aspx

This one allows for either as well and confirms that your total is correct for permutation with repetition and my without

http://keisan.casio.com/exec/system/1223625156

It also shows that the Combination with repetition is 3,190,187,286
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by:d-glitch
ID: 41824207
If you have n choices for the first position, and (n-1) for the second . . .
Then the correct formula will be using factorial.

If you have n choices for the first position, and n for the second . . .
Then factorial will be incorrect.
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Expert Comment

by:pony10us
ID: 41824245
Well - good discussion as usual and I concede that your answer is correct based on the criteria set forth in the question.  :)
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