modulomathy.com

#math trivia: The five classes eat lunch at separate times. Every day their times are in a different order. How long can this go on?

— Burt Kaliski Jr. (@modulomathy) September 17, 2012

There are 5*4*3*2*1 = 120 different possible orders in which the five classes can eat lunch, so this can go on for 120 days.

The rationale is as follows:  Any one of the five classes can eat lunch first. With that class having eaten, there are four classes that could eat second, then three that could eat third, and so on.  Multiplying the number of possibilities at each stage together gives the total number of different orders.

Beyond the trivia:  The number of different possible orders or permutations of n items is n*(n-1)*(n-2)*…*1, which is denoted n! for short, and called n factorial. Factorial numbers are helpful in solving many problems in combinatorics. They also grow very quickly. The first 10 factorial numbers are: