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The "Pity Counter" resets when one pulls a 5-star item (character or weapon). When the 5-star is not the last in the 10 rolls, does the pity counter start from 0 after the 10-pull or (0 < n < 10)? For example, assume I pull Zhongli on his banner as the first of the 10 rolls. Is the pity after the 10-pull 0 or 9?

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Each pity counter resets on the exact pull that awarded you the 4* / 5* item. In your example, if you do a 10 pull multi and get a 5* character as the fourth item, the remaining six pulls are counted towards your next pity.

This means that it is perfectly equivalent to do a single 10 multi pull or 10 separate ones. The only advantages are time spent in the first case and the ability to stop earlier in the second one. Short version: always do single pulls, that way you can stop anytime you get the item you want and will waste less fate should you pull on the wrong banner by mistake. Also, most of the time you don't want to buy the fates with primos in the Paimon bargain shop - just try to wish and the game will prompt you to buy the correct type of fate, preventing you from buying the wrong one by mistake.

Notice that the wording on the event wish rules page never mention 10 pulls multi, so doing anything different from the above would be false advertising and actionable. As far as I know contrary to popular stereotypes, China has actually far more rigid rules regarding gacha rates than many other countries so that is probably a big no.

Anyway, since you asked, let's add some more clarifications here:

You are guaranteed a featured 4* item every 20 pulls
Every time you get a 4* item, there is a 50% (75% on the weapon banner) chance it will be one of the featured items for the current banner. If you don't get a featured item, the next one will be.

Event characters have a 50% rate, event weapons 75%
When getting a 5* drop in an event banner, you have a 50%/75% probability to get one of the featured event items. If you get Qiqi instead don't get the featured item, your next pity is guaranteed to be the event item. This means that you are guaranteed an event item every 180 pulls (or 160 on the weapon banner)

The 4*/10 pulls and 5*/90pulls pity counter are probably separate.
The wish rules state that

"Every 10 wishes is guaranteed to include at least one 4 star or higher item".

This seems to indicate that getting a 5* should reset the 4* counter. Yet, empirical evidence seems to prove that if you get one 5* item on your tenth pull towards a 4* item, the 4* item is "delayed" and will be forced as your next pull. Notice that if this is indeed true it would not violate the rule above - the 5* item counts as a "4 star or higher item"

There is concrete evidence of the existence of a soft pity
While the rules simply state that there is a pity at 90 pulls for a 5* item (80 for weapons) empirical evidence again shows that there is a soft pity starting at 74 pulls. This means that starting on the 74th pull the probability to get a 5* item increases with every pull up to 100% on the 90th one (again, numbers are different on the weapon banner but you get the idea). This is why some not-so-wise players think it is wise to "build up pity" up to 73 on banners they don't want the 5* item from... and end up with the wrong item, wasting their pity in the process.

For a final summary, please refer to this picture courtesy of genshin-wishes.

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