Find The Rank Of A Word

Python 3, 71 bytes

lambda s:sum(t<=(*s,)for t in{*permutations(s)})
from itertools import*

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05AB1E, 5 bytes

ϐsk>

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Explanation

œ       # get permutations of input
 ê      # sort and remove duplicates
  s     # swap input to top of stack
   k    # get index of input in the list
    >   # increment

Gaia, 4 bytes

fuȯI

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