how many switches do i need to connect 200 computers

That depends on the workload (e.g. how many are servers, how many are workstations). For example, arithmetically, fifteen sixteen-port switches are needed; one central into which fourteen others are attached, and then the two hundred workstations plug into those. This may not be the optimal arrangement for performance, however. If groups of systems share certain workloads, you'll want to isolate them to their own switch or switches, to keep their traffic from slowing down other systems.
 
See that depends where your pc are located. If they are all very close to each other then you can use one big one with lots of 48 port blades in it or you can use stackable switches that "appear" to be a single large switch and use special cable to interconnect the backplanes.

If your PC are spread out then you need to know how many groups of PC you have and what there traffic patterns are. For example if you had 2 buildings you could put a switch in each and maybe cable them together with a 10g network.

 

dbhosttexas

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Jan 15, 2013
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How are all the computers distributed?

At a minimum, 5.

5 @ 48 ports = 225 ports - 8 for interconnection = 217 ports.

You will want switches with AT LEAST 2 10GbaseT ports for uplinking between each other.

This is not exactly an optimized approach however.