# Number of Lock Combinations

• Oct 14th 2009, 07:33 PM
centenial
Number of Lock Combinations
This puzzle has me scratching my head:

Quote:

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!
• Oct 14th 2009, 08:10 PM
aman_cc
Quote:

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