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The apartment complex has some potential liability, but I am afraid that you leaving the faucet running was the primary cause, so you would be primarily liable for the damages. As far as the clogged pipes, you can sue in court and have the apartment complex found liable for some share, the percentage is up to the judge, but typically it would be about 20%-30% liability in this case, because BUT FOR you leaving the faucet running unattended and unsupervised, the flooding would not have occurred even if the pipes were clogged because you would have caught it if you were paying attention.
So even if you prove it was the pipes, you would not escape primary responsibility, because you were liable to pay attention to the faucet while it was running as that is a duty of a tenant.
Thank you for your reply.
Quite honestly, you would spend money fighting this but in the end, you are likely to be held liable for the damage. But that does not mean you should not send a letter to the apartment and tell them that you are offering to pay 1/2 because you believe 1/2 is their fault because of the pipes and try to negotiate a lesser amount. In the long run though, if you go to court, they will ultimately hold you liable.