SOLUTION: Can you please help me with this? A modern Master combination lock is electronic and the combination consists of a series of up, down, left, and right clicks. The user-created

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Question 908836: Can you please help me with this?
A modern Master combination lock is electronic and the combination consists of a series of up, down, left, and right clicks. The user-created combination can vary from 4-12 clicks. Again, order is important, and repetition is allowed. How many combinations are possible with a 4-click combination?
P = (n*4)r = (12*4)4 = 484 = 5,308,416 is what I tried.
Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!
[By the way, the word "combination" when used to refer to a combination
lock has nothing to do with the idea of "combinations" in math courses.]  

If I'm interpreting your problem correctly, you might have a 
lock combination of:

4 clicks, for instance Up,Down.Down,Right, abbreviated UDDR.
There would be 4*4*4*4 or 44 such lock combinations.

Or you might have a lock combination with 5 clicks, for instance, 
Left, Left, Right, Down, UP, abbreviated LLRDU,
There would be 4*4*4*4*4 or 45 such lock combinations.

or you might have a lock combination with any number of clicks up
to and including 12 clicks.  For instance if you had a lock 
combination with 12 clicks, it  might be DDRRRULRLLUU.
There would be 412 such lock combinations.

Since there are 4 clicks, if you have a string on n clicks there 
would be 4*4*4*...*4 to n factors or 4n possible 
combinations consisting of n clicks..

Therefore the total number of lock combinations would be 

44+45+46+47+48+49+410+411+412

which is the sum of a geometric series with , , and 

A formula for the sum of a geometric series is


 
Edwin
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