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
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
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".
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 $n$.