ArPen vs AD
I can't make sense of the math. I see no difference between picking AD over ArPen. This is the way I understand it using small round numbers:
I do 3 points of attack damage. Enemy has no armor. Each of my basic attacks to 3 points of damage. Enemy picks up 3 points of armor. I do 3 points of AD Enemy has 3 points of armor. Each of my basic attacks to 0 points of damage. So I have two options. Take AD or ArPen. I take 3 points of ArPen. I do 3 points of AD. Enemy has 3 armor. But I have 3 points of ArPen canceling out his 3 armor. Each of my basic attacks to 3 points of damage. OR I take 3 more points of AD. Now I do 6 points of AD. Enemy has 3 points of armor. Each of my basic attacks to 3 points of damage. So taking AD or ArPen result in the same amount of damage. But, I don't think this is correct. Can anyone help me understand the advantage of one over the other and/or help me understand the true math behind the stats? 
Playing as an ADC (and one of my favorite roles), you should focus on building up raw damage first, then armor penetration. Personally I start trying to rush Infinity Edge after obtaining some of my early items (doran's blade, boots, and vampiric specter). For armor penetration, I'd go for the Last Whisper, as it deals more armor penetration than Black Cleaver.
There are cases, however, that the enemy team tend to build early armor (particularly the bruiser type characters) and I still want to have good damage output. In this case, I'd rush Black Cleaver first then Phantom Dancer/Infinity Edge 
Well first off, that's not how armor mitigates damage. But even if it were, that's still not how you decide between AD and ArmPen. Can you last hit 100% of the minions without AD runes? If not, run AD runes.

Thank you for the suggested builds, however, I'm trying to learn the math. When you say "That's not how armor mitigates damage", what do you mean?
Please focus on my understanding of ArPen and AD, and armor for that matter as apposed to builds. I would like to understand the math so I can decide on my own what the best build is! 
Here's a good video that discusses about armor penetration and items between Black Cleaver and Last Whisper. He cites both advantages and disadvantages of each item:
http://www.youtube.com/watch?v=_7P09AVK3yE 
Oh, I was talking about runes, not builds. You need to have AD flat runes so you can last hit all the minions. This will help you have more gold to build whatever items it is that you want to build.
But with that said, armor works as a damage multiplier. The math goes like this: Incoming physical damage is multiplied by a factor based on the unit's armor: Damage multiplier = 100 / (100 + Armor) if Armor ≥ 0 Damage multiplier = 2 − 100 / (100 − Armor) if Armor ≤ 0 And then "Armor" is whatever amount of armor the target has after your penetration and reduction. Armor reduction is calculated before percentage penetrations. So if the enemy has 100 armor, and you've got say 20 flat reduction and 50% armor penetration, then the enemy has: 100  20 = 80 80  ( 80 * 0.50 ) = 40 armor. Then we put the 40 armor into the damage multiplier equation: 100 / ( 100 + 40 ) = 100 / 140 = 0.714. And then we multiply that by whatever damage your physical attack would do (whether autoattack or an ability that does physical damage), so if your autoattack damage is 100, then: 100 * 0.714 = 71.4 damage per autoattack to this target. By increasing armor penetration or reduction, you increase your damage multiplier. 
Well there are two types of armor penetration (flat and %) the reason people get last whisper to get kills is because it has % penetration and is better vs people with high amounts of armor.
Flat penetration is more effective vs low armor targets whereas % penetration is better vs high armor targets. Now to your question: early game getting AD is better because enemies dont have as much armor and you are mostly hitting minions for gold ( its easier to last hit with more AD). Most AD players get AD (IE or BT) then ASpeed(phantom dancer) then armor % penetration(LW) As for your calculations they are mostly correct and prove my point about early AD you generally do about the same damage and its easier to last hit.However late game the calcution will be a bit skewed when the tanks on their team will have 200+ armor. To be able to kill them fast AD is not enough and you need armor % pene. Also BC is worse than LW because it requires a lot of auto attacks to be as effective as lw. 
The key to me understanding this was finding out that I had armor all wrong! Thanks for pointing me in the right direction and for the explanation.

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