Question

A group of 20 friends are going to travel to a school reunion together, 7 of them can drive. They have a 7 seat car (1 driver and 6 passengers), a 5 seat car (1 driver and 4 passengers) and the remaining 8 will take a bus. How many ways can the friends travel (ignoring passenger seat positions within each vehicle)? Give an exact answer, do not round. Do not include commas or dots in your answer.

Answer 1

There are 7 friends who can drive, and they can distribute themselves among the available cars in 7 ways. The remaining friends will take the bus, resulting in a total of 105 ways that the friends can travel.

Step-by-step explanation:

To find the number of ways the friends can travel, we need to consider the different possibilities for each friend. There are 7 friends who can drive, so we need to determine how these 7 friends can distribute themselves among the available cars.

Let's consider the first car with 7 seats. The driver will take one seat, leaving 6 remaining seats. Since there are 7 friends who can drive, we can choose any combination of 6 friends to fill the remaining seats in the car. This can be calculated using the combination formula:

Combination = C(7, 6) = 7! / (6! * (7-6)!) = 7 ways

Next, let's consider the second car with 5 seats. The process is similar, except now we have 6 friends left and 4 seats to fill. This can be calculated using the combination formula:

Combination = C(6, 4) = 6! / (4! * (6-4)!) = 15 ways

Finally, the remaining 8 friends will take the bus. There is only one bus available, so the number of possibilities is 1.

To find the total number of ways the friends can travel, we multiply the number of possibilities for each mode of transportation:

Total possibilities = 7 * 15 * 1 = 105 ways

Answer 2

Step-by-step explanation:

7 can drive, so let us select the driving seats in the cars first:

We select one of the drivers for the first car: 7C1

⇒ We are left with 6 drivers who may be selected for the second car: 6C1

For the remaining seats, the first car has 6 seats left: 18C6 (We already selected 2 from 20 - the drivers)

In the second car it is 12C4 (Subtracting the 6 people taken into the first car, and there being for more seats in the second car)

The rest of the friends will have to go in the bus, 8 of them - there is only one way to do this because 8C8 = 1

∴ The answer is 7C1 · 6C1 · 18C6 · 12C4 · 8C8 = 385945560