Question 40

We break each of the links in the circled triple to form $3$ open links. Use one of them to join B and C, one of them to join D and E, and the third to join A and F.

There have been $3$ breaks and $3$ rewelds, with a total cost of $3\times$ $$1+3\times$ $$2=$ $$9$

The possible patterns are:

• $5$ on one wire and $0$ on the other three wires. There are $4$ of these.
• $4$ on one wire, $1$ on one wire, and $0$ on the other two wires. There are $4\times3=12$ of these.
• $3$ on one wire, $2$ on one wire, and $0$ on the other two wires. There are $4\times3=12$ of these.
• $3$ on one wire, $1$ on two wires, and $0$ on the other wire. There are $4\times3=12$ of these.
• $2$ on two wires, $1$ on one wire, and $0$ on the other wire. There are $4\times3=12$ of these.
• $2$ on one wire and $1$ on three other wires. There are $4$ of these.

Thus, the total number of different patterns is $2\times4+4\times12=56$.