Final answer:
The probability that you and your friend will be seated together at a round table with five chairs is 1/5. This is determined by considering the pair as a single entity and analyzing the possible distinct seating arrangements considering the rotation of the table.
Step-by-step explanation:
To determine the probability that you and your friend will be seated together at a round table with five chairs, we can use a simple combinatorial approach. We consider that once you take a seat, there are four remaining seats, with two of them being next to you. Thus, your friend has a 2 in 4 chance to sit next to you if seated randomly. However, because the table is round, every seating arrangement can be rotated to look like any other, effectively making the probability simpler to calculate. You take a seat, which leaves two adjacent seats for your friend. We can treat the combination of you and your friend as a single entity, which leads us to find combinations of seating this entity around the table.
There are 5 possible positions for this pair (you and your friend) to be seated together at the table. Once the pair is seated, the rest of the people can sit in any order. Since the pair and the other three are all seated together, there are 5! (120) total arrangements of seating everyone, but we divide by the number of ways we can rotate the table (5), which is akin to fixing one person's position and arranging the rest. Therefore, the total number of distinct seating arrangements is 24. Out of these, there are 4! (24) arrangements where you and your friend are not together (again we fix your position and arrange the others). So, the probability is 1 - 4!/5! = 1 - 1/5 = 4/5. However, these are the ways you are not together, and the probability of being together is the complement of this which is 1/5.
The correct answer is a) {1/5}.