A group of twelve people are going to take a ride on a roller coaster at the same time in two different trains, one of which cannot fit more than 8 passengers, and the other cannot fit more than 7. In how many ways can the group take a ride?
3003
To find the number of ways the group can take a ride, we need to consider the different possibilities for the number of people in each train.
Case 1: 8 people in the first train and 4 people in the second train.
In this case, we have C(12, 8) ways to select the 8 people for the first train and C(4, 4) ways to select the 4 people for the second train. Here, C(n, r) represents the number of combinations of n items taken r at a time.
Case 2: 7 people in the first train and 5 people in the second train.
In this case, we have C(12, 7) ways to select the 7 people for the first train and C(5, 5) ways to select the 5 people for the second train.
Case 3: 6 people in the first train and 6 people in the second train.
In this case, we have C(12, 6) ways to select the 6 people for the first train and C(6, 6) ways to select the 6 people for the second train.
Now, we can calculate the total number of ways by summing up the number of ways from each case:
Total number of ways = C(12, 8) * C(4, 4) + C(12, 7) * C(5, 5) + C(12, 6) * C(6, 6)
Using the formula for combinations: C(n, r) = n! / (r! * (n-r)!)
= 12! / (8! * 4!) * 4! / (4! * 0!) + 12! / (7! * 5!) * 5! / (5! * 0!) + 12! / (6! * 6!) * 6! / (6! * 0!)
= 495 * 1 + 792 * 1 + 924 * 1
= 495 + 792 + 924
= 2211
Therefore, the group can take a ride in 2211 different ways.
To find the number of ways the group can take a ride, we can use the concept of permutations.
First, let's consider the train that can fit a maximum of 8 passengers. We need to determine the number of ways we can select at most 8 people from the group of 12 people.
To do this, we can consider different possibilities:
1. If all 12 people choose to ride on the train that can fit 8 passengers, we have only one possible combination.
2. If 11 people choose to ride on the train that can fit 8 passengers, we have 12 possible choices for the person who will ride on the other train, which can fit a maximum of 7 passengers.
3. If 10 people choose to ride on the train that can fit 8 passengers, we have 12 possible choices for the two people who will ride on the other train.
4. If 9 people choose to ride on the train that can fit 8 passengers, we have 12 possible choices for the three people who will ride on the other train.
5. If 8 people choose to ride on the train that can fit 8 passengers, we have 12 possible choices for the four people who will ride on the other train.
In total, there are 12 choices for the number of people who ride on the train that can fit 8 passengers, ranging from 8 people to all 12 people. For each choice, there are specific combinations of people who can ride on the other train.
Now, let's consider the train that can fit a maximum of 7 passengers. For each combination of people who ride on the train that can fit 8 passengers, we need to determine the number of ways to select at most 7 people from the remaining people who did not ride on the first train.
Using the same approach as before, we can calculate the number of choices and combinations:
1. If all remaining 4 people choose to ride on the train that can fit 7 passengers, we have only one possible combination.
2. If 3 people choose to ride on the train that can fit 7 passengers, we have 4 possible choices for the person who will not ride on either train.
3. If 2 people choose to ride on the train that can fit 7 passengers, we have 6 possible choices for the two people who will not ride on either train.
4. If 1 person chooses to ride on the train that can fit 7 passengers, we have 4 possible choices for the three people who will not ride on either train.
5. If no one chooses to ride on the train that can fit 7 passengers, we have 5 possible choices for the four people who will not ride on either train.
Finally, we can multiply the number of choices for each train together to find the total number of ways the group can take a ride:
Choices for train 1 (up to 8 people) * Choices for train 2 (up to 7 people)
Summing up all the combinations, we get:
(1 + 12 + 12 + 12 + 12) * (1 + 4 + 6 + 4 + 5) = 60 * 20 = 1200
Therefore, there are a total of 1200 ways the group of twelve people can take a ride on the roller coaster.