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How is this called:

$\frac{3!}{2!} + \frac{3!}{1!} + \frac{3!}{0!} = 15$

For example with a,b,c it would be:

a,ab,ac,abc,acb,

b,ba,bc,bac,bca,

c,ca,cb,cab,cba

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    Are you referring to _permutation_?2012-01-03

2 Answers 2

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You might be prepared to extend it to 3!/3! + 3!/2! + 3!/1! + 3!/0! = 16 to include the case where you have an empty ordered subset.

More generally for positive $n$ $\sum_{k=0}^n \frac{n!}{k!} = \lfloor n! \times e \rfloor.$

OEIS A000522 and A007526 suggest these are called "arrangements", but centuries ago were called "variations".

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It sounds like you're referring to counting.

$\sum_{k=1}^n \frac{n!}{k!}$ tells you the number of nonempty tuples obtainable from a set of size $n$.