Is there any data in Tolkien (books/letters/etc..) to explain **why there were precisely 9 Men rings and 7 Dwarf rings and 3 Elf rings**? (<sub>the latter assumes that Sauron planned the nmber of Elvish rings to be 3 before he was thwarted by Elves making the rings themselves</sub>)

Was it because of the amount of tribes/kings he needed to deal with? Some magical/power limitations? Magical properties of numbers 9 and 7 (ala 7 Horcruxes)? A whim of Sauron's?

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**UPDATE**: Looks like 3 and 7 are explained in the answers (3 elf lords and supposedly 7 Dwarf houses, though nobody provided canon proof).

***What was NOT explained was why there were 9 Man rings?*** 

 - If 9 was a **meaningful amount, why**?

 - If it was merely "19-7-3" leftover, **then why were there 19 total non-One rings made**?