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Verified Answer :

I believe the answer would be 4!
So 4 3 2 1
24.

Here's proof :
ABCD
ABDC
ACBD
ACDB
ADBC
ADCB
BACD
BADC
BCAD
BCDA
BDAC
BDCA

Following this order, each letter has 6 additional options, so 4! Would answer this appropriately. Some would say 6
4 but that's the same as 432 (3 and 2 becomes 6)