2 A team with five members is training to take part in a quiz. The team members, Abby, Bob, Cait, Drew and Ellie, attempted sample questions on each of the five topics and their scores are given in the table. For the actual quiz, each topic must be allocated to exactly one of the team members. The maximum total score for the sample questions is to be used to allocate the different topics to the team members.

1. Explain why the Hungarian algorithm may be used if each number, \(x\), in the table is replaced by \(35 - x\).
2. Form a new table by subtracting each number in the table above from 35 . Hence show that, by reducing rows first then columns, the resulting table of values is as below, stating the values of the constants \(p\) and \(q\).

   8	6	8	0	4
   0	11	\(p\)	4	4
   10	4	6	0	12
   \(q\)	2	0	4	0
   0	0	6	6	0

3. Show that the zeros in the table in part (b) can be covered with two horizontal and two vertical lines. Hence use the Hungarian algorithm to reduce the table to a form where five lines are needed to cover the zeros.
4. 1. Hence find the possible allocations of topics to the five team members so that the total score for the sample questions is maximised.
   2. State the value of this maximum total score.