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Math Help - Number of Lock Combinations

  1. #1
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    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!
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  2. #2
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    Quote Originally Posted by centenial View Post
    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
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