Final answer:
To meet the network design requirements of 50 subnets with 200 hosts per subnet, a Class B network should use a subnet mask of 255.255.252.0, which allows for 64 subnets and 1022 usable IP addresses, thus fulfilling both requirements.
Step-by-step explanation:
The network design calls for 50 subnets with a maximum of 200 hosts per subnet within a Class B network. In a Class B network, by default, the first 16 bits are used for the network address and the remaining 16 bits for host addresses. To support at least 200 hosts, you would need enough host bits to accommodate that number, which is 8 bits (28 = 256); however, 2 addresses are unusable (network and broadcast), so you actually have 254 usable addresses, which satisfies the need for 200 hosts.
In this case, to create 50 subnets, you would require at least a 6-bit subnet (26= 64 subnets possible), thus leaving 10 bits for hosts. Therefore, the appropriate subnet mask would be 255.255.252.0 (/22 in CIDR notation), which provides 64 subnets and 1022 usable IP addresses per subnet (210 - 2).
To maximize the number of subnets, you would use as many bits for subnetting as possible without going below 200 hosts per subnet. As mentioned, the 255.255.252.0 mask would give you the most subnets while still maintaining the necessary number of hosts. To maximize the number of hosts, you could use a subnet mask like 255.255.255.0 (/24 in CIDR notation), which would give you 254 usable IP addresses but only allow for 256 subnets.