# Thread: Number of Lock Combinations

1. ## Number of Lock Combinations

This puzzle has me scratching my head:

A special type of door lock has a panel with five buttons labeled with the digits 1 through 5. This lock is opened by a sequence of three actions. Each action consists of either pressing one of the buttons or pressing a pair of them simultaneously. For example, 12-4-3 is a possible combination. The combination 12-4-3 is the same as 21-4-3 because both the 12 and the 21 simply mean to press buttons 1 and 2 simultaneously.

a. How many combinations are possible?
b. Hom many combinations are possible if no digit is repeated in the combination?
I started out by isolating the 'single digit' and 'multiple digit' combinations. If you entertain only single digit actions, there are 5^3 possibilities. Likewise, if you entertain only multiple digit actions, there are 5!^3 possibilities. So, there are 5^3*5!^3 total possibilities for part a. Is that right?

I'm not sure where to start with part b. Any hints would be greatly appreciated!

2. Originally Posted by centenial
This puzzle has me scratching my head:

I started out by isolating the 'single digit' and 'multiple digit' combinations. If you entertain only single digit actions, there are 5^3 possibilities. Likewise, if you entertain only multiple digit actions, there are 5!^3 possibilities. So, there are 5^3*5!^3 total possibilities for part a. Is that right?

I'm not sure where to start with part b. Any hints would be greatly appreciated!
Hi for your part a. I am getting
(5C1+5C2)^3

This is with repetition allowed.

For restricting that I trust you will have to take cases like
1-1-1
2-1-1
1-2-1
1-1-2
2-2-1
2-1-2
1-2-2