Personally I found running many games at once does not work for me, I have gotten cards when it was 3 or less games at once.
Cards generally drop once every 30 minutes for most people when playing a single game I am unsure if this average stays the same when using multiple games.
I do not believe there is a limit to how many you can obtain in a day since there is a limit already associated to the total amount you can have.
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