# How is this combinatorial entity called?

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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I would call it "how many nonempty strings can be made from the letters $\{a,b,c\}$ without using any letter more than once". –  Henning Makholm Jan 3 '12 at 0:19
Are you referring to permutation? –  Sniper Clown Jan 3 '12 at 0:23

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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+1 for references. –  matt Jan 3 '12 at 2:51

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 $n$.

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