This is going to boil down to who is at fault for this flooding.
If the faucets were broken and were not functioning properly prior to this overflow and you have brought it to the attention of the landlord and he/she failed to fix it, then generally the landlord would be fount to be at fault.
Also there is no requirement that a bathtub needs to be installed with an overflow safety, as many older buildings/houses still have old fashioned bathtubs.
If the bathtub was clogged through no fault of your own, it would normally be the responsibility of the landlord to fix it, but you had to provide him a notice of this problem.
Unfortunately you might potentially be found to be at least partially liable for the damage and I would suggest that you consult with a local real estate
attorney to protect your legal rights and interests.
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