Any odd ductwork or other holes/obstacles involved, or do you have a clear field both between joist cavities and below them?
How much too loud is your situation as is?
What kind of noise are you most concerned with - footfalls, TV noise, kids going berserk, etc -
What size room, area of the ceiling, and budget?
The GOOD news is, you only need to lose about 1 3/4" of headroom and use pretty inexpensive materials to do this, barring any wierd obstacles you didn't mention - also, you've already found the one material that can be tricky to find, the rest are easy.
You will likely need to lose about 3-4 more inches of headroom directly over the mix position for acoustic treatment, and I wouldn't recommend carpet (possibly a small area rug though) but either vinyl or laminate would be fine.
If you can answer the above questions, we can get started... Steve