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Consider that $15$ machines need to be connected in a LAN using $8$-port Ethernet switches. Assume that these switches do not have any separate uplink ports. The minimum number of switches needed is ______
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Answer is $3.$ 

Using $3$ switches we can connect maximum $20$ machines together.

∴ We require at least $3$ switches to connect $15$ machines.

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There are 15 devices. And there are 8 ports in 1 Ethernet switch. So it's obvious that one switch is not sufficient. So one more switch is required. So in 1st switch only 7 ports are connected to 7 devices and 8th one is used for connecting to another switch. Now come to second switch. Only 7 ports are left as 1st port is used for connecting to 1st switch. Those 7 ports can be connected to 7 devices. But there are total 15 devices and so 1 would be left. So we need another switch. So now, in second switch we connect 6 devices to 6 ports and last port to third switch. And in the last switch 1st port is connected to 2nd switch and two ports to two left devices.So we require 3 switches in total.
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Given :- A switch with $8$ ports and we require to connect $15$ machines together.

Using $1$ switch we can connect only $8$ machines together.

 

Using $2$ switches we can connect only $14$ machines together.

 

Using $3$ switches we can connect $20$ machines together.

 

$\therefore$ We require atleast $3$ switches to connect $15$ machines together.

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