Mathematics · Ch 5 — Permutations and Combinations
Introduction
Introduction
The Opening Problem: A Number Lock
Suppose you have a suitcase secured by a number lock with 4 wheels, each showing the digits 0 through 9. The lock opens only when the four wheels are set to one specific sequence, with no digit repeated. You have forgotten this sequence — except you remember that the first digit is 7.
To open the lock, you would need to try different sequences for the remaining 3 wheels until you find the right one. How many different 3-digit sequences might you have to check?
You could try to answer this by listing every possible arrangement of the 9 remaining digits (0 to 9, excluding 7), taken 3 at a time. But that would be slow and impractical — the number of possible sequences is large enough that listing them by hand is not realistic.
This chapter exists to answer exactly this kind of question. Instead of listing every possibility one by one, it develops basic counting techniques that let us find how many arrangements or selections are possible — directly, without writing any of them out.
Where This Is Heading
These counting techniques are useful whenever we need to know the number of different ways of arranging objects (where the order in which they are placed matters, as with the digits on the lock) or selecting objects (where only which ones are chosen matters, not their order) — without actually listing every possibility.
As a first step toward building these techniques, the next section examines a principle that is fundamental to all of them.