2021 ASRJC P2 Q7

Four friends went for a Christmas party, each bringing a present for gift exchange.During the gift exchange, the presents were randomly distributed, such that each person received exactly one gift, which may be the same or different from the present that he brought along.It is given that the presents brought to the Christmas party were all distinct from one another.

1. Find the total number of ways to distribute the four presents to the four people if there are no restrictions.[1]
2. Find the number of ways the presents can be distributed if there is exactly one person who received back their own presents.[2]
3. Find the number of ways the presents can be distributed give that there are exactly two persons who received back his own present.[2]
4. Explain why there cannot be a case where there are exactly \(3\) persons who received back their own gifts.[1]
5. Hence or otherwise, find the probability that none of the friends received back their own present after the gift exchange.[2]