three students are to be seated in a row of 10 chairs. What is the probability that they will be seated in adjacent seats
There are 10! ways to place 10 students in 10 chairs in a row.
There are 8! ways to arrange a group of 3 students with 7 other students.
Among the three students (in a group), they can arrange themselves in 3! different ways.
So the probability that they will be seated together is
(8!3!)/10!
To find the probability that the three students will be seated in adjacent seats, we need to determine the total number of possible seating arrangements and the number of arrangements where the students are seated together.
Step 1: Determine the total number of possible seating arrangements.
Since there are 10 chairs and three students, we can assign each student to a chair one by one. For the first student, we have 10 choices, for the second student we have 9 choices, and for the third student, we have 8 choices remaining. Therefore, the total number of seating arrangements is given by:
10 x 9 x 8 = 720
Step 2: Determine the number of arrangements where the students are seated together.
Since the students need to be seated together, we can treat them as a single unit. This leaves us with 8 remaining seats. We can assign these 8 seats to the group of students in (8 choose 1) ways. Additionally, within the group of students, there are 3 ways the students can be arranged. Therefore, the number of seating arrangements where the students are together is given by:
8 x 3 = 24
Step 3: Calculate the probability.
To find the probability, we divide the number of arrangements where the students are seated together by the total number of seating arrangements:
Probability = Number of arrangements where the students are seated together / Total number of seating arrangements
Probability = 24 / 720
Probability = 1 / 30
So, the probability that the three students will be seated in adjacent seats is 1/30 or approximately 0.0333.