#math trivia: The principal chooses one student from each of 5 classes of 20 for a project team. How many different “rosters” are possible?
— Burt Kaliski Jr. (@modulomathy) October 1, 2012
The number of possible rosters is 3.2 million. This assumes that it doesn’t matter in what order the students are chosen for the project. There are 20 possible students who can represent the first class, 20 who can represent the second, and so on. The choices for the five classes are independent of one another, so the number of possibilities can be multiplied together:
20*20*20*20*20 = 3,200,000 .
If order mattered, then the number possible rosters would be 120 times larger: The student representing the first class could occupy one of five positions, the one from the second class one of the four remaining positions, etc.; 5*4*3*2*1 = 120. The total number would be
3,200,000 * 120 = 384,000, 000 .
Beyond the trivia: Suppose that instead of choosing one student per class, the principal simply choose five students from the school overall? The number of possible rosters would then be (if order mattered):
100*99*98*97*96 = 9,034,502,400 .
If order didn’t matter then the number would be 120 times smaller:
9,034,502,400 / 120 = 75,287,520 .
Note that this number is still much larger than if the students had to be from different classes. Just consider the last student chosen to get a sense of the difference. If students are chosen by class, there are only 20 possibilities for that last choice. If they are chosen from the school, there are still 96 remaining to choose from after four have been selected because they can be drawn from all five classes. Fewer constraints always means more possibilities.