Minecraft – What enchantments can bows have

To update: Replace all instances of level 50 with level 30, redo all the math, and treat this answer as more updated than it actually is.

In Short, yes, getting to level 50 gives you the best chances of multiple high-end enchantments.

In Long:

You're kind of riding a grumpy horse on this one. You want the most efficient level to get enchantments. That would be level 50, there is no ifs, ands, or buts about it. Flat out, a higher level gives you the greatest chances of getting multiple enchants. Period.

You can still get the highest level of specific enchants at some levels lower than 50 depending on the type of item you enchant, but generally speaking, I would strongly suggest you enchant diamond tools\weapons\armor only, because they last longer so you get more bang for your 'buck' of experience. As listed on the wiki, Diamond Items don't have the best qualities for enchanting, but 1,300+ uses from a Pickaxe of Fortune III is just too good to pass up if you ask me.

Multiple Enchantments

The formulae for multiple enchants works like this:

1--After the first element is selected, there is a chance of receiving more, based on this algorithm:

2--Divide the modified level in half, rounded down. (This does not affect the possible enchantments themselves, because they were all pre-calculated in Step Two.)

3--With probability (modified level + 1) / 50, keep going. Otherwise, stop picking bonus enchantments.

4--Remove from the list of possible enchantments anything that conflicts with previously-chosen enchantments.

5--Pick one enchantment from the remaining possible enchantments (based on the weights, as before) and apply it to the item.

6--Repeat from step 1.

Now what this means for you is that the higher your MODIFIED LEVEL is during your enchantment instance, the greater chance you have for the keep going. And those two words are what gives an item multiple enchantments.

In the second step of the process, you can see that it divides your modified level in half. I will go into what a modified level is later in the post, but let's just say it's your actual level (50), for the sake of argument. Now divide 50 in half for twenty five, add one to it (26) and throw it on top of /50. That gives you just a bit better than a 50\50 shot on two enchants for your weapon. That's not bad. It gets much much worse as you keep going though. Each time the respective chances are halved. Rather than do all the math here on Gaming.SE the wiki states:

A level 50 diamond pickaxe would have a 16.10% chance to have 3 enchantments, a 48.09% chance to have exactly 2 enchantments and a 35.80% chance to have exactly 1 enchantment.

You can see there that at level 50 you actually have a Greater chance to get two enchantments than you do to get one. And only 16.10% chance to get three. If you want to get three your best chances are at level 50, and they are still not very good.

Specific Enchantments by Level

This part is somewhat less tricky because the wiki provides us with a wonderfully easy to read graph. I however, suck at Microsoft Paint, and cannot make a sexy.jpg of it. I'll do my best here.
(Modified Enchantment Level in parentheses)

Efficiency Enchantment
Level 1 (1-51) Level 2 (15-66) Level 3 (33-81) Level 4 (46-96) Level 5 (61-111)

Silk Touch Enchantment
Level 1 (25-75)

Unbreaking Enchantment
Level 1 (5-55) Level 2 (15-65) Level 3 (25-75)

Fortune Enchantment
Level 1 (20-70) Level 2 (32-82) Level 3 (44-94)

This section is pretty straightforward. If your modified enchanting level falls into any of these number brackets you have the possibility to get that enchantment.

Some notes:
This part is decided before the number of enchantments are decided. So the halving of your MEL (modified enchantment level) doesn't affect this adversely. All possible enchants are decided before the # of enchantments comes into play.

If your MEL falls into TWO different level brackets of the same enchantment, then the highest possible level is used.

You cannot get Silk Touch and Fortune. Getting one automatically disqualifies your item for the other.

Modified Enchantment Level

All that stated, you're going to need to know how to get your MEL. Which is easy.

modified enchantment level = enchantment level + Random(0, enchantability) + 1

Enchantment Level = Your Experience Level

Random(0,enchantability) + 1 = a random number between 0 and the items enchantibility, and then add 1 to that.

(enchantibility is an items enchantibility determined by the type of item it is, diamond, gold, wood, etc. These are listed on the wiki as well.)

Once you have your MEL from the above formulae, the game modifies it further one last time, either increasing or reducing it by a random percentage that caps at 25% more or less than it was originally. (100 can turn into anything from 75 to 125)

I think this is everything you would need to know. Let me know if I missed something in comments and I'll be sure to add it in or clarify it.

After a significant amount of testing, of which GLaDOS would be proud, we managed to come up with a fairly simple table of data from which to formulate an answer.

^{Variation of about .5 blocks}

To reach these results, I first enchanted my weapons until I had Knockback I and Knockback II (note: the minimum modified levels for these enchants are 5 and 25, respectively).

I then had my associate spawn Zombies in the Knockback Testing Super System™ so that I could attack the Zombies from a safe, secure location.

We first tested a Normal Sword (no enchantment) from a standing position, and then again while sprinting. We then repeated the process with enchanted swords. Once we reached a definitive conclusion, we moved on to testing with Bows and their related enchantments (in this case Punch). The results of our SCIENCE can be found at the top of this answer.

I hope this helped!