Question

2. A train consists of an engine at the front, a caboose at the rear, and 27 boxcars that are numbered from 1 to 27.

a. How many different orders are there for cars that make up the train?
b. If the cars are attached to the train in a random order, what is the probability that the boxcars are in numerical
order from 1 to 27?

Answer 1

a)
`n = 27! = 27*26*25.......*3*2*1= 1.109x10^(28)`

b)
`p = (1)/(27!) = (1)/(1.109x10^(28))=9.18x10^(-29)`

Explanation:

Previous concepts

The rule of product or multiplication principle is a basic counting principle thats is used in order to find the number of ways to do anything. If we assume that we have a ways to do one thing and b ways to do other thing in total we have a*b to do both things.

Part a

For this case we can use the multiplication principle and in order to find the number of ways that we can order the cars in order to construc the train would be:

`n = 27! = 27*26*25.......*3*2*1= 1.109x10^(28)`

Part b

For this case we want that the boxcars would be in order from 1 to 27 and thats 1 of the possibility out of the
`1.109x10^(28)` others so then the probability would be given by:

`p = (1)/(27!) = (1)/(1.109x10^(28))=9.18x10^(-29)`