Possible combinations with thirteen buttons

davidbdix

This is similar to one of my previous posts, but is quite different.
There are 13 buttons (numbered 1, 2, 3, .....13)in this combination lock, and you have to figure out which ones can be pressed in.
They can either be pushed in or left into their original extended position.
You don't know how many buttons need to be pushed in (or left pushed out) to solve the lock (it could be none, 1, 3, 8, or all 13).
How many possibile combinations are there? Any how do you arrive at that answer?

Here's a photo of my lock. The buttons are in the front of the blanket chest and are spring-loaded. When you push one of the buttons, the corresponding lock-pin on the top of the chest falls into place due to gravity. To let the lock return to the original position, you pull the lock-pin upwards and the spring forces the button back out. After each attempt at pushing the buttons, the longer bar on top of the chest gets pushed down. If the combination is correct, you can push the bar all the way down. If incorrect, it only depressed part of the way.

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tiny-tim

hi davidbdix!

have you tried labelling each button's position as either 0 or 1 ?

mathman

(I answered on another forum). 2¹³. Expand (1+1)¹³ as binomial. The kth term is the number of ways to press k buttons.

davidbdix

thanks!