Name: Ken

Who is asking: Other
Level: All

Question:
What is total number of combinations possible on a combination lock which contains 3 wheels each numbered 0-9?

This is a nice situation to use to introduce permutations. (Don't be confused by the language. The lock is called a combination lock and the "code" that opens the lock is called a combination, but you are actually counting permutations.)

Suppose that you are going to open the lock by trying all possible combinations in an orderly fashion. One way is to set the first two wheels to 0 and then move the third wheel through all ten values. That is try 000, 001, 002..., 009, ten in all. If none of these work, leave the first wheel at 0, set the second wheel to 1 and move the third wheel through all ten values, 010, 011, 012..., 019. Contunue on, set the first wheel to 0, the second wheel to 2 and move the third wheel through all ten values, etc.

What you see is that if the first wheel is set to 0, then for each of the ten possible positions of the second wheel there are ten choices for the third wheel. Thus if the first wheel is set to 0 there are 10 x 10 = 100 different combinations.

Now set the first wheel to 1 and repeat the process. Again you get 100 different combinations, each with the first wheel remaining at 1. Repeat for the first wheel set to 2, then 3, etc., each time getting 100 new combinations. Thus for each of the ten possible settings for the first wheel ther are 100 different combinations. Hence in total there are 10 x 100 = 1000 different combinations.