Find the least number of cables required to connect eight computers to four printers to guarantee that for every choice of four of the eight computers, these four computers can directly access four different printers. Justify your answer.

We will need or require at least 20 cables to connect eight computers to four printer to guarantee that for every choice of four of the eight computers, these four computers can directly access four different printers

Step-by-step explanation:

To start with solving this question, Let the computers be labelled C1, C2, C3, C4, C5, C6, C7, C8 and the printers be P1, P2, P3, P4.

If we now choose to connect connect Ci to Pi, for 1 ≤ i ≤ 4( that is for one less or equal to i less or equal to 4), and connect Cj to all printers, for 5 ≤ j ≤ 8 ( five less or equal to j less or equal to 8 ), we will discover that we will need:

4 + ( 4 x 4 ) = 20 cables.

We can then conclude that this connection satisfies the condition that any four computers can directly be connected or access four different printers.

To Justify this, we will go ahead to prove thus:

We will show it is impossible to use less than 20 cables. Assume 19 cables are used to connect eight computers to four printers. The average number of computers connected to a printer is 19 / 4 which is less than 5.

Therefore, some printer must be connected to less than five computers, thus, this printer is connected to four or less than 4 computers.

Then, there are at least four computers not connected to it. Hence, it is not possible for these four computers to access four printers because they are directly connected to at most three printers.

So referencing back to the question, we can conclude that we will need at least 20 cables.