1 At the end of each year the workers at an office take part in a gift exchange.
Each worker randomly chooses the name of one other worker and buys a small gift for that person.
Each worker's name is chosen by exactly one of the others.
A worker cannot choose their own name.
In the first year there were four workers, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D .
There are 9 ways in which A, B, C and D can choose the names for the gift exchange. One of these is already given in the table in the Printed Answer Booklet.
- Complete the table in the Printed Answer Booklet to show the remaining 8 ways in which the names can be chosen.
During the second year, worker D left and was replaced with worker E.
The organiser of the gift exchange wants to know whether it is possible for the event to happen for another 3 years (starting with the second year) with none of the workers choosing a name they have chosen before, assuming that there are no further changes in the workers. - Classify the organiser's problem as an existence, construction, enumeration or optimisation problem.
After the second year, the organiser drew a graph showing who each worker chose in the first two years of the gift exchange.
None of the workers chose the same name in the first and second years.
The vertices of the graph represented the workers, A, B, C, D and E, and the arcs showed who had been chosen by each worker. - Explain why the graph must be a digraph.
- State the number of arcs in the digraph that shows the choices for the first two years.
- Assuming that the digraph created in part (d) is planar, use Euler's formula to calculate how many regions it has.