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The Canonical Starting Point: Factorials

Almost every textbook and lecture that teaches recursion starts with factorials. That really is a fine way to introduce the technique, so we won't buck tradition today.

The factorial of a positive integer is calculated by multiplying it by all positive integers smaller than itself. (Factorials are undefined for negative numbers, and the factorial of 0 is defined to be 1.) Standard mathematical notation for the factorial of a number n is n!, so we say, 5! = 1 × 2 × 3 × 4 × 5 = 120.

Factorials are often used for problems of probability and counting. For instance, how many ways can a standard deck of cards be shuffled? For each of the 52 cards that can end up on the bottom, there are 51 possibilities for ...