How many possible guesses? 
A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $ 1$ to $ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were $1, 1, 1, 1, 3, 3, 3$. Find the total number of possible guesses for all three prizes consistent with the hint.

This is a confusing problem, I think casework is required. 
If Case 1:
$$A-4; B-2, 1; C-1, 2$$
So if $A$ has a 4digit value then $B$ has a $2$ or $1$ and $C$ has a $1$ or $2$. 
$$1, 1, 1, 1, 3, 3, 3$$ 
But that would take way too long. I tried stars and bars.
But that gave me too low of an answer, so I am not sure where to turn next.  
edit: I tried the casework didn't go too well.
I lined it up to:
$$\binom{7}{4} [\binom{3}{2} + \binom{3}{1}] 2 + \binom{7}{3}[\binom{4}{3} + \binom{4}{2} + \binom{4}{1}][3] + ... + \binom{7}{1}[...][5]$$
It have an overlarge answer.
What to do here?
HINTS ONLY PLEASE!
 A: Not sure if hint, more like an advice: Thinking about it with numbers might make it more confusing than it really is. For example:

Your contestant is given 4 red balls and 3 green balls. He must organize them in 3 ordered stacks A,B,C. No less than 1 ball per group, and no more than 4. How many ways are there to distribute them?

You can easily pretend the given problem is this because with those numbers you'll never find a situation where a 4 digit number goes over your limit.
A: Hint: There are ${7 \choose 3}$ ways to order the digits, basically by choosing where the 3's go. Then you want to split the ordered digits into 3 contiguous groups (think stars and bars) such that each group has length at least 1, and you rule out the case where 2 groups have only length 1 (say, by brute force counting) because then you would have a group with 5 digits which isn't allowed.
A: Hint: First count the distinct orderings of the digits (e.g. 1331131). For each such ordering, count the number of ways to insert two commas (e.g. 1,331,131) at least one space but no more than 4 spaces apart.
