Final answer:
The probabilities for different seating arrangements for 9 students assigned randomly are calculated using basic probability and the concept of factorial (for arranging all in designated seats) and derangement (for arranging none in designated seats).
Step-by-step explanation:
The question at hand deals with probability and specifically with assigning seats to students at random.
Answer to the Provided Question
- a. The probability that the first student sits in a designated seat is simply 1/9 since there are 9 students and 9 designated seats.
- b. The probability that all students sit in their designated seats is 1/9! (1 divided by 9 factorial), because each student has one specific place, and it reduces the number of possible correct choices for each next student.
- c. The probability that no student sits in their designated seat is a case of a derangement, where no element appears in its original position. This is also known as the subfactorial of 9, which can be calculated using the formula !n.
- d. Each student sitting in a different random seat is a certainty, as random distribution by definition ensures that all students will indeed sit in random seats, albeit whether those seats are designated or not is not defined in this scenario. If the query implies random seats different from their own, this is also a case of derangement as in part c.