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Math: Number of possible combinations when sorted alphabetically

Posted on 2014-02-03
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Last Modified: 2014-02-03
I have 17 items:
ABCDEFGHIJKLMNOPQ
I am working on a project that combines PDF documents in order, and I need to know how many total options would be possible.

There can be any combination of them, providing each item can only appear in the list once and the results are always in alphabetical order.

For example these would all be valid,
ACB
ACE
BFJ

But these would not be valid:
BBB
BCA

How can the number of options be calculated?
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Question by:hankknight
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by:John-Charles-Herzberg
John-Charles-Herzberg earned 20 total points
ID: 39829948
Combinations without repetition (n=17, r=3)

List has 680 entries.
{a,b,c} {a,b,d} {a,b,e} {a,b,f} {a,b,g} {a,b,h} {a,b,i} {a,b,j} {a,b,k} {a,b,l} {a,b,m} {a,b,n} {a,b,o} {a,b,p} {a,b,q} {a,c,d} {a,c,e} {a,c,f} {a,c,g} {a,c,h} {a,c,i} {a,c,j} {a,c,k} {a,c,l} {a,c,m} {a,c,n} {a,c,o} {a,c,p} {a,c,q} {a,d,e} {a,d,f} {a,d,g} {a,d,h} {a,d,i} {a,d,j} {a,d,k} {a,d,l} {a,d,m} {a,d,n} {a,d,o} {a,d,p} {a,d,q} {a,e,f} {a,e,g} {a,e,h} {a,e,i} {a,e,j} {a,e,k} {a,e,l} {a,e,m} {a,e,n} {a,e,o} {a,e,p} {a,e,q} {a,f,g} {a,f,h} {a,f,i} {a,f,j} {a,f,k} {a,f,l} {a,f,m} {a,f,n} {a,f,o} {a,f,p} {a,f,q} {a,g,h} {a,g,i} {a,g,j} {a,g,k} {a,g,l} {a,g,m} {a,g,n} {a,g,o} {a,g,p} {a,g,q} {a,h,i} {a,h,j} {a,h,k} {a,h,l} {a,h,m} {a,h,n} {a,h,o} {a,h,p} {a,h,q} {a,i,j} {a,i,k} {a,i,l} {a,i,m} {a,i,n} {a,i,o} {a,i,p} {a,i,q} {a,j,k} {a,j,l} {a,j,m} {a,j,n} {a,j,o} {a,j,p} {a,j,q} {a,k,l} {a,k,m} {a,k,n} {a,k,o} {a,k,p} {a,k,q} {a,l,m} {a,l,n} {a,l,o} {a,l,p} {a,l,q} {a,m,n} {a,m,o} {a,m,p} {a,m,q} {a,n,o} {a,n,p} {a,n,q} {a,o,p} {a,o,q} {a,p,q} {b,c,d} {b,c,e} {b,c,f} {b,c,g} {b,c,h} {b,c,i} {b,c,j} {b,c,k} {b,c,l} {b,c,m} {b,c,n} {b,c,o} {b,c,p} {b,c,q} {b,d,e} {b,d,f} {b,d,g} {b,d,h} {b,d,i} {b,d,j} {b,d,k} {b,d,l} {b,d,m} {b,d,n} {b,d,o} {b,d,p} {b,d,q} {b,e,f} {b,e,g} {b,e,h} {b,e,i} {b,e,j} {b,e,k} {b,e,l} {b,e,m} {b,e,n} {b,e,o} {b,e,p} {b,e,q} {b,f,g} {b,f,h} {b,f,i} {b,f,j} {b,f,k} {b,f,l} {b,f,m} {b,f,n} {b,f,o} {b,f,p} {b,f,q} {b,g,h} {b,g,i} {b,g,j} {b,g,k} {b,g,l} {b,g,m} {b,g,n} {b,g,o} {b,g,p} {b,g,q} {b,h,i} {b,h,j} {b,h,k} {b,h,l} {b,h,m} {b,h,n} {b,h,o} {b,h,p} {b,h,q} {b,i,j} {b,i,k} {b,i,l} {b,i,m} {b,i,n} {b,i,o} {b,i,p} {b,i,q} {b,j,k} {b,j,l} {b,j,m} {b,j,n} {b,j,o} {b,j,p} {b,j,q} {b,k,l} {b,k,m} {b,k,n} {b,k,o} {b,k,p} {b,k,q} {b,l,m} {b,l,n} {b,l,o} {b,l,p} {b,l,q} {b,m,n} {b,m,o} {b,m,p} {b,m,q} {b,n,o} {b,n,p} {b,n,q} {b,o,p} {b,o,q} {b,p,q} {c,d,e} {c,d,f} {c,d,g} {c,d,h} {c,d,i} {c,d,j} {c,d,k} {c,d,l} {c,d,m} {c,d,n} {c,d,o} {c,d,p} {c,d,q} {c,e,f} {c,e,g} {c,e,h} {c,e,i} {c,e,j} {c,e,k} {c,e,l} {c,e,m} {c,e,n} {c,e,o} {c,e,p} {c,e,q} {c,f,g} {c,f,h} {c,f,i} {c,f,j} {c,f,k} {c,f,l} {c,f,m} {c,f,n} {c,f,o} {c,f,p} {c,f,q} {c,g,h} {c,g,i} {c,g,j} {c,g,k} {c,g,l} {c,g,m} {c,g,n} {c,g,o} {c,g,p} {c,g,q} {c,h,i} {c,h,j} {c,h,k} {c,h,l} {c,h,m} {c,h,n} {c,h,o} {c,h,p} {c,h,q} {c,i,j} {c,i,k} {c,i,l} {c,i,m} {c,i,n} {c,i,o} {c,i,p} {c,i,q} {c,j,k} {c,j,l} {c,j,m} {c,j,n} {c,j,o} {c,j,p} {c,j,q} {c,k,l} {c,k,m} {c,k,n} {c,k,o} {c,k,p} {c,k,q} {c,l,m} {c,l,n} {c,l,o} {c,l,p} {c,l,q} {c,m,n} {c,m,o} {c,m,p} {c,m,q} {c,n,o} {c,n,p} {c,n,q} {c,o,p} {c,o,q} {c,p,q} {d,e,f} {d,e,g} {d,e,h} {d,e,i} {d,e,j} {d,e,k} {d,e,l} {d,e,m} {d,e,n} {d,e,o} {d,e,p} {d,e,q} {d,f,g} {d,f,h} {d,f,i} {d,f,j} {d,f,k} {d,f,l} {d,f,m} {d,f,n} {d,f,o} {d,f,p} {d,f,q} {d,g,h} {d,g,i} {d,g,j} {d,g,k} {d,g,l} {d,g,m} {d,g,n} {d,g,o} {d,g,p} {d,g,q} {d,h,i} {d,h,j} {d,h,k} {d,h,l} {d,h,m} {d,h,n} {d,h,o} {d,h,p} {d,h,q} {d,i,j} {d,i,k} {d,i,l} {d,i,m} {d,i,n} {d,i,o} {d,i,p} {d,i,q} {d,j,k} {d,j,l} {d,j,m} {d,j,n} {d,j,o} {d,j,p} {d,j,q} {d,k,l} {d,k,m} {d,k,n} {d,k,o} {d,k,p} {d,k,q} {d,l,m} {d,l,n} {d,l,o} {d,l,p} {d,l,q} {d,m,n} {d,m,o} {d,m,p} {d,m,q} {d,n,o} {d,n,p} {d,n,q} {d,o,p} {d,o,q} {d,p,q} {e,f,g} {e,f,h} {e,f,i} {e,f,j} {e,f,k} {e,f,l} {e,f,m} {e,f,n} {e,f,o} {e,f,p} {e,f,q} {e,g,h} {e,g,i} {e,g,j} {e,g,k} {e,g,l} {e,g,m} {e,g,n} {e,g,o} {e,g,p} {e,g,q} {e,h,i} {e,h,j} {e,h,k} {e,h,l} {e,h,m} {e,h,n} {e,h,o} {e,h,p} {e,h,q} {e,i,j} {e,i,k} {e,i,l} {e,i,m} {e,i,n} {e,i,o} {e,i,p} {e,i,q} {e,j,k} {e,j,l} {e,j,m} {e,j,n} {e,j,o} {e,j,p} {e,j,q} {e,k,l} {e,k,m} {e,k,n} {e,k,o} {e,k,p} {e,k,q} {e,l,m} {e,l,n} {e,l,o} {e,l,p} {e,l,q} {e,m,n} {e,m,o} {e,m,p} {e,m,q} {e,n,o} {e,n,p} {e,n,q} {e,o,p} {e,o,q} {e,p,q} {f,g,h} {f,g,i} {f,g,j} {f,g,k} {f,g,l} {f,g,m} {f,g,n} {f,g,o} {f,g,p} {f,g,q} {f,h,i} {f,h,j} {f,h,k} {f,h,l} {f,h,m} {f,h,n} {f,h,o} {f,h,p} {f,h,q} {f,i,j} {f,i,k} {f,i,l} {f,i,m} {f,i,n} {f,i,o} {f,i,p} {f,i,q} {f,j,k} {f,j,l} {f,j,m} {f,j,n} {f,j,o} {f,j,p} {f,j,q} {f,k,l} {f,k,m} {f,k,n} {f,k,o} {f,k,p} {f,k,q} {f,l,m} {f,l,n} {f,l,o} {f,l,p} {f,l,q} {f,m,n} {f,m,o} {f,m,p} {f,m,q} {f,n,o} {f,n,p} {f,n,q} {f,o,p} {f,o,q} {f,p,q} {g,h,i} {g,h,j} {g,h,k} {g,h,l} {g,h,m} {g,h,n} {g,h,o} {g,h,p} {g,h,q} {g,i,j} {g,i,k} {g,i,l} {g,i,m} {g,i,n} {g,i,o} {g,i,p} {g,i,q} {g,j,k} {g,j,l} {g,j,m} {g,j,n} {g,j,o} {g,j,p} {g,j,q} {g,k,l} {g,k,m} {g,k,n} {g,k,o} {g,k,p} {g,k,q} {g,l,m} {g,l,n} {g,l,o} {g,l,p} {g,l,q} {g,m,n} {g,m,o} {g,m,p} {g,m,q} {g,n,o} {g,n,p} {g,n,q} {g,o,p} {g,o,q} {g,p,q} {h,i,j} {h,i,k} {h,i,l} {h,i,m} {h,i,n} {h,i,o} {h,i,p} {h,i,q} {h,j,k} {h,j,l} {h,j,m} {h,j,n} {h,j,o} {h,j,p} {h,j,q} {h,k,l} {h,k,m} {h,k,n} {h,k,o} {h,k,p} {h,k,q} {h,l,m} {h,l,n} {h,l,o} {h,l,p} {h,l,q} {h,m,n} {h,m,o} {h,m,p} {h,m,q} {h,n,o} {h,n,p} {h,n,q} {h,o,p} {h,o,q} {h,p,q} {i,j,k} {i,j,l} {i,j,m} {i,j,n} {i,j,o} {i,j,p} {i,j,q} {i,k,l} {i,k,m} {i,k,n} {i,k,o} {i,k,p} {i,k,q} {i,l,m} {i,l,n} {i,l,o} {i,l,p} {i,l,q} {i,m,n} {i,m,o} {i,m,p} {i,m,q} {i,n,o} {i,n,p} {i,n,q} {i,o,p} {i,o,q} {i,p,q} {j,k,l} {j,k,m} {j,k,n} {j,k,o} {j,k,p} {j,k,q} {j,l,m} {j,l,n} {j,l,o} {j,l,p} {j,l,q} {j,m,n} {j,m,o} {j,m,p} {j,m,q} {j,n,o} {j,n,p} {j,n,q} {j,o,p} {j,o,q} {j,p,q} {k,l,m} {k,l,n} {k,l,o} {k,l,p} {k,l,q} {k,m,n} {k,m,o} {k,m,p} {k,m,q} {k,n,o} {k,n,p} {k,n,q} {k,o,p} {k,o,q} {k,p,q} {l,m,n} {l,m,o} {l,m,p} {l,m,q} {l,n,o} {l,n,p} {l,n,q} {l,o,p} {l,o,q} {l,p,q} {m,n,o} {m,n,p} {m,n,q} {m,o,p} {m,o,q} {m,p,q} {n,o,p} {n,o,q} {n,p,q} {o,p,q}

If you need more information.  The calculator can be found at:

Combinations and Permutations Calculator

http://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html

Thanks
JC
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Assisted Solution

by:ozo
ozo earned 20 total points
ID: 39829985
17!/(17-3)!/3!
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Accepted Solution

by:
phoffric earned 360 total points
ID: 39830000
>> the results are always in alphabetical order.
>> ACB
   Why is this allowed? Shouldn't it be ABC?

If you want options that are larger than 3 (or even 2, or just 1), then the total number of options is 2^17 less 1. To see why, take a binary word that is 17 bits long. If you select ABDE, then that could map to:
11011000000000000

To select all options, you would select all possibilities of 1's and 0's. There are 2^17 such possibilities. One of those possibilities is all 0's which means none are selected. That is why the total number of options is 2^17 less 1.
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Assisted Solution

by:TommySzalapski
TommySzalapski earned 100 total points
ID: 39830004
Are you always selecting three?
If you can select any number, than the total possible combinations is
2^17 (which includes the case where you select none and the one where you select all)
So valid cases
[blank]
A
B
AB
ACDG
GHPQ
ABCDEFGHIJKLMNOPQ
etc

Oops phoffric beat me to the punch.
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Author Comment

by:hankknight
ID: 39830028
Thank you-- 2 to the power of 17 minus one is the answer I was looking for.
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