Which one is correct/appropriate to ask
How many vowels are there in the English alphabet?
How many vowels are there in the English Language?
Or are they both correct/appropriate?
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Both are appropriate, but they ask two different things.
How many vowels are there in the English alphabet? asks about the characters in the writing system.
But as I'm sure you're aware, English has far more vowel sounds than characters to represent them. How many vowels are there in the English language? asks about the distinctive sounds—phonemes—in the spoken language.
The first one is wrong because "alphabet" should be singular. There is only one English alphabet.
If you fix this, and use alphabet instead of alphabets then they are both correct, and work just fine. I would prefer the first one, since vowels are more directly related to the alphabet.