Combinatorics

A powerful application of enumeration to computers and algorithms is the recurrence relation. A sequence of numbers can be generated using the previous numbers in a sequence by using a recurrence relation. This recurrence relation either adds to, or multiplies one or more previous elements of the sequence to generate the next sequence number. The factorial, n!, is solved using a recurrence relation since n! equals n × (n-1)! and (n-1)! equals (n-1) × (n-2)! and so on. Eventually one factorial is reached, which is just one. Pascal's triangle is also a recurrence relation. Computers, being based on algorithms, are designed to calculate and count numbers in this way.