Solve Travel Problem: 10 Fans in 3 Taxis

[Gotcha]

Here is a problem that I need to solve in preparation of my exams
(I know how to use combinations, permutations, binomial coeeficcients, normal distributions):

Ten fans are traveling to a rugby match in three taxis that will hold 2, 4 and 5 passengers respectively. In how many ways is it possible to transport the 10 people to the rugby match.

 

[ColdRifle]

1st 2 in taxi1 4 in Taxi2 and 4 in taxi3
2nd 2 in Taxi1 5 in taxi3 and 3 in Taxi2
3rd 1 in taxi1 5in taxi3 and 4 in taxi2
so there are 3 possible ways...thats because you have only one spare seat in all three taxis so the spare seat can be in all three taxis

 

[tacman]

To solve this travel problem, we can use combinations and permutations to determine the different ways in which the 10 fans can be transported to the rugby match using the three taxis.

First, we can use combinations to determine the number of ways in which the 10 fans can be divided into groups of 2, 4, and 5. This can be calculated as follows:

C(10,2) = 45 ways to choose 2 fans from the group of 10
C(8,4) = 70 ways to choose 4 fans from the remaining group of 8
C(4,5) = 1 way to choose all 5 fans from the remaining group of 4

Therefore, the total number of ways to divide the 10 fans into groups of 2, 4, and 5 is 45 x 70 x 1 = 3,150 ways.

Next, we can use permutations to determine the number of ways in which the groups of fans can be assigned to the three taxis. This can be calculated as follows:

P(3,3) = 6 ways to assign the groups of fans to the three taxis

Finally, the total number of ways in which the 10 fans can be transported to the rugby match is the product of the number of ways to divide the fans into groups and the number of ways to assign the groups to taxis. Therefore, the total number of ways is 3,150 x 6 = 18,900 ways.

Alternatively, we can also use the binomial coefficient to solve this problem. The number of ways to transport the 10 fans can be calculated as follows:

C(10,2) x C(8,4) x C(4,5) x P(3,3)
= (10!/(2!(10-2)!)) x (8!/(4!(8-4)!)) x (4!/(5!(4-5)!)) x (3!/(3!(3-3)!))
= (45) x (70) x (1) x (6)
= 18,900 ways

Lastly, we can also use the normal distribution to solve this problem. The number of ways to transport the 10 fans can be approximated by using the central limit theorem, which states that for a large sample size, the sampling distribution of the sample means is approximately normal. In this case, we can consider