A data center is used by a large cloud service provider for hosting critical applications. The center has $15,000$ servers, each with a Mean Time To Failure $($MTTF$)$ of $60$ days. The entire system will only experience catastrophic failure if $25\%$ or more of the servers become non$-$operational.

a$.$ What is the MTTF for the system, assuming that the failure of each server is independent?

b$.$ If it costs an additional $$3,000$ per server to increase the MTTF to $120$ days, would it be a cost$-$effective solution? Justify your answer by calculating the new system MTTF and comparing it to the increased cost.