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A company makes a -button combination lock. The buttons on this lock are labeled 1,2 and 3. Combinations for this lock follow these rules:

A combination is a sequence of one or more pushes, each of which involve at least one button. When two or more buttons are pushed simultaneously, the order they're pushed in doesn't matter.

Each button may be used at most once, and combinations do not need to include all buttons.


For example, one possible combination for this lock is to push button 3 and then push buttons 1 and 2 simultaneously. Another possible combination for this lock is to just push buttons 2 and 3 simultaneously.

How many possible combinations does this combination lock have?

 Jun 6, 2023
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There are 8 possible combinations for this combination lock.

To find the number of possible combinations, we can consider each button individually. There are 3 choices for each button, so there are 33=27 possible combinations if we consider all possible combinations of buttons. However, some of these combinations are the same, since the order in which we push the buttons doesn't matter. For example, pushing buttons 1 and 2 is the same as pushing buttons 2 and 1. So, we need to divide the number of possible combinations by the number of ways to order the buttons. There are 3!=6 ways to order 3 buttons, so there are 627​=8 possible combinations.

Here is a table of all 8 possible combinations:

 

Button 1 | Button 2 | Button 3 | Combination

------- | -------- | -------- | --------

No push | No push | No push

No push | No push | 1 | 1

No push | No push | 2 | 2

No push | No push | 3 | 3

No push | 1 | 2 | 12

No push | 1 | 3 | 13

No push | 2 | 3 | 23

1 | 2 | 3 | 123

 Jun 7, 2023

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