It's not that simple. Each pack you open has 5 slots, each of those 5 slots can be anything from common to legendary (I think one "blue or better" is guaranteed though?). Duplicate protection means that you won't get the same card again until you have all others of that rarity. But, it does NOT mean that if you have 100% commons, that your packs won't roll a common slot.
So suppose you have 100% of the common, legendary, and rare cards from the set, and no epics at all. You can still open a pack that is common, common, common, common, rare. Duplicate protection doesn't guarantee you get the missing cards, you are just guaranteed to get no duplicate epics IF your pack rolls one more more epic slots.
edit: TLDR there is no exact number of packs required. It is probably hundreds less than it used to be before duplicate protection, but even when you have all the common and rares, you have a low chance of getting packs with epics and legendaries.
The duplicate protections doesn't change the rate of occurence for the different rarities. So to get ALL of the cards, it won't be fewer than before. Legendaries are still the limiting factor here, and nothing has changed for them!
It is now just much more likely that you will get a decent collection of rares and even epics with fewer packs. You will never open a third copy of one of those, as long as there are other cards of the same rarity that you DON'T have two of yet. So that will certainly help. It just won't make getting ALL of the cards (which includes legendaries) easier.
That's completely wrong. There are different drop rates for each rarity, you get an epic on average every 4 packs and a legendary on average every 20 packs after getting 1 in your first 10 packs. So if you want to get everything from just packs then you would need around 480 but of course you would get a tonne of dust from duplicates so you can just craft the stuff you're missing hence i'd say somewhere around the 180-200 mark to get the whole collection depending on your luck.
How many threads do we need of this? The drop rates aren't changing, it's still going to take a lot of packs to get all the legendaries but you will get more different epics and rare and commons because of the no duplicate rule.
To do it strictly through opening likely around 200, but after about 40 packs you’ll get dupes for commons and you’ll probably start getting rare dupes around 60 packs so if you factor in dust probably about 150-170 packs. Or, if you’re like me and dust everything that goes to wild, the preorder mega bundle will be plenty. I’m still sitting on 36k dust after the HoF rotation and some of that is left over from the last rotation.