Number of possible arrangements of rings on a hand This is a homework question that I'm having trouble figuring out how to start.
Here's the question. A woman has 3 different rings. On any given day she wears 1, 2, or (inclusive) 3 of her rings on her hand (but never wears any ring on her thumb). How many different ways can she wear her rings? Different arrangements of multiple rings on a single finger count as different.
My initial intuition is to break this into subcases, i.e. she wears 1 ring, 2 rings, 3 rings, etc. but I'm not sure where to proceed from there. I believe for 1 ring it's as simple as (but I'm still not entirely sure)
$${4 \choose 1}{3 \choose 1} = 12$$
As for background, this is for a 'foundations of mathematics' course with no textbook. I'm currently a senior in college studying computer science with a minor in mathematics so this course is required. I've taken a course in discrete mathematics where combinatorics was covered, but not to this degree and I've never really been 'good' with combinatorics so very ground level pointers would be appreciated.
 A: Let the rings be labeled $a,b,c$.  Introduce an additional symbol, $\mid$, which will be used to denote on which finger we are placing said rings.
$\underbrace{~~~~~~}_{\text{pointer finger}}\mid\underbrace{~~~~~}_{\text{middle finger}} \mid \underbrace{a}_{\text{ring finger}}\mid\underbrace{cb}_{\text{pinky finger}}$
The space to the left of the furthest left $\mid$ will represent the pointer finger, inbetween the first and second $\mid$'s the middle finger, etc...  Further, if multiple rings are placed in a region, we consider the top-down arrangement on the finger the same as the left-right arrangement in the line.
So, the arrangement above, written more compactly as "$\mid\mid a\mid cb$" represents the pointer and middle fingers empty, ring $a$ on the ring finger, and rings $c$ and $b$ in that order on the pinky finger.
Check your understanding by interpreting the following arrangements of symbols as arrangements of rings on the hand:  
$c\mid a\mid\mid b$
$abc\mid\mid\mid$
$\mid abc\mid\mid$

How many ways can we arrange the symbols $a,b,c,\mid,\mid,\mid$?  Why is this useful and what does it count in relation to our scenario?

 There are $\binom{6}{1,1,1,3} = \frac{6!}{1!1!1!3!} = 6\cdot 5\cdot 4 = 120$ arrangements of the symbols and this can be seen as a way of counting the number of ways that she wears all three rings simultaneously.


For counting how many ways she wears exactly one or exactly two rings, first pick which one ring or which two rings are being used, and then use a similar technique as before, now with fewer symbols involved.
For example, with one ring, we pick which of $a,b,c$ is being used.  We then arrange $x,\mid,\mid,\mid$.  This gives $3\cdot \binom{4}{1,3}=3\cdot \binom{4}{1}=12$ as you found earlier.

As an aside, this technique is often referred to as stars and bars and can be applied to several related problems.
A: With $n$ rings, $k$ of them used, and $4$ fingers, we can generalize the formula to
$ \binom{k+3}{3}\times^nP_k $
The first term is placement of rings treated as identical, using stars and bars
and the second term is permutations of the rings which are actually distinct. With $n= 3:$
$k = 1:\;\; \binom43\times 3 = 12$
$k = 2:\;\; \binom53\times 3\cdot2 = 60$
$k = 3:\;\; \binom63\times 3\cdot2\cdot1 = 120$  
