A train goes one-way to 20 stations and at each station picks up a group of people, one of whom will get off at each of the remaining stations. For example, at station 5, the train picks up a group of people. One person of the group will get off at station 6, another at station 7, another at station 8, and so forth. At station 20, the last person of that group will get off.
a) What is the total number of people that will get on the train?
b) What is the greatest number of people that will be on the train at any station? Assume that the number of people who get on and off of the train at any station do so at the same instant.
(Source: Mathematics Teaching in the Middle School, Oct 1995)