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I recently saw someone use '200.1.2.0/18.' This doesn't make any sense to me. This is a Class C address with a subnet mask of /18. I thought the subnet mask for a Class C had to be at least /24. How exactly does this work?

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That address uses what's called Classless Inter-Domain Routing (or CIDR for short). The class system has largely been abolished due to the rapidly diminishing IPv4 space.