Asked in Math and Arithmetic, Statistics, Algebra
Math and Arithmetic
Statistics
Algebra

How can you find the total number of possible combinations for a combination lock? 07/28/2010

First of all, even though we call them "combination locks" they are not combination locks. In fact, there is no such thing as a "combination lock", that is if you pay homage to the actual, technical meaning of the word "combination". The word combination implies that order is irrelevant, which is not the case on a combination lock. The numbers for the "combination" have a particular order, and that order makes the difference between the lock opening or not. Just because you get the numbers correct doesn't mean the lock will open unless you get them in the right order, too. A combination lock is more appropriately called a permutation lock.

Now, you want to know the number of permutations for a combination lock. This is a much more precise question.

The next thing we need to know is whether or not repeats are allowed. In combination locks it is generally the case that repeats are allowed. One number does not effect the next number. Not only do the number of possibilities remain constant for each number, but which values they can possess also remains constant.

Say for example we have a combination lock with three numbers to be set. Suppose that each of those can be the numbers 0 to 29. In this case, if the sequence 5, 6, 3 is distinct from 6, 3, 5, then we have a permutation lock. Not a combination lock. If the two sequences were the same, if both opened the lock... then its a true combination lock indeed. If the lock can open to 5, 5, 3 then repeats are allowed.

This lock has 30 unique values for each number in the sequence: 0-29. And it has 3 numbers in the sequence. If repeats are allowed then there are 303 = 27,000 permutations. If repeats are not allowed then there are 30P3 = 24,360 permutations.