There are 20 bikes in spinning class, the bikes are arranged 4 rows, with 5 bikes in each row. They hope to be in the same row, but cannot request specific bike. Determine the probability that all 5 friends will be in same row with Jeff and dariya at either end.
At the ends it could be Jeff, Dariya or Dariya, Jeff
So for the first row:
2x3x2x1 or 12 ways, leaving 15! for the other positions.
but there are 4 rows,
prob(of your event) = 4x12(15!)/20! = .000025799
To determine the probability that all 5 friends will be in the same row with Jeff and Dariya at either end, we need to calculate the total number of possible arrangements where this condition is met, and divide it by the total number of possible arrangements.
Firstly, let's consider the number of ways to arrange Jeff and Dariya within a row. Since they must be at either end, there are only 2 available positions for them.
Next, we need to place the remaining 5 friends within the same row. There are 4 rows available, and we want them to be in the same row. Since the arrangement of friends is irrelevant, we can choose any row to place them in.
Therefore, the total number of favorable arrangements is 2 (for the positions of Jeff and Dariya) multiplied by 4 (available row choices) which equals 8.
To calculate the total number of possible arrangements, we consider all 20 bikes. The first bike can be placed in any of the 20 positions, the second bike can be placed in any of the remaining 19 positions, and so on, until the last bike has only one available position. So, the total number of possible arrangements is given by 20 × 19 × 18 × 17 × 16.
Lastly, to determine the probability, we divide the number of favorable arrangements by the total number of possible arrangements:
Probability = (Number of favorable arrangements) / (Total number of possible arrangements)
= 8 / (20 × 19 × 18 × 17 × 16)
To simplify this expression, you can calculate the values and obtain the probability in decimal or fraction form.