three students are to be seated in a row of 10 chairs. What is the probability that they will be seated in adjacent seats

how many possible ways can three students sit in ten chairs?

10!/(10-3)!

10!/7!

10*9*8

This makes sense b/c there are 10 spots for the first students, 9 spots for the second student and 8 spots for the third.

How many ways can 3 students sit adjacent?
123
234
345
456
567
678
789
8910
but remember that these are unique people so they can sit in different orders (or 3! ways three people can sit in 3 seats, 6)

so (8*6)/(10*9*8)
6/90
3/45
1/15

1/15 chance of having 3 students sitting adjacently in a row of 10 chairs.

To calculate the probability of three students being seated in adjacent seats in a row of 10 chairs, we need to determine the total number of possible seating arrangements where the students are adjacent, and then divide it by the total number of possible seating arrangements.

To find the total number of possible seating arrangements where the students are adjacent, we can consider the three students as a single entity or block. Therefore, we can treat them as a group and calculate the number of arrangements of the remaining 7 chairs and the group of students.

Since we are treating the three students as a single block, we have 8 entities to arrange (the block of students and the remaining 7 chairs), which can be done in 8! = 40,320 ways.

Next, let's calculate the total number of possible seating arrangements without any restrictions. We have 10 chairs and 3 students, which can be arranged in 10! = 3,628,800 ways.

Finally, to find the probability, we divide the number of favorable outcomes (seating arrangements with the students adjacent) by the total number of possible outcomes (all seating arrangements).

Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 40,320 / 3,628,800

We can simplify the calculation further by dividing the numerator and denominator by their greatest common divisor (gcd). In this case, both numbers are divisible by 720.

Probability = (40,320 ÷ 720) / (3,628,800 ÷ 720)
Probability = 56 / 5,040
Probability = 1 / 90

Therefore, the probability that the three students will be seated in adjacent seats is 1/90 or approximately 0.0111.