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.

 

[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!

 

[blue_raver22]

I would approach this problem using combinatorics, which is the branch of mathematics that deals with counting and organizing combinations and permutations.

To determine the number of possible combinations with thirteen buttons, we can use the formula for combinations, which is nCr = n!/(r!(n-r)!), where n is the total number of objects and r is the number of objects being selected. In this case, n = 13 (total number of buttons) and r can range from 0 (none of the buttons being pushed in) to 13 (all buttons being pushed in).

Therefore, the total number of possible combinations is the sum of all possible combinations for r = 0 to 13, which can be written as:

nCr = 13C0 + 13C1 + 13C2 + 13C3 + 13C4 + 13C5 + 13C6 + 13C7 + 13C8 + 13C9 + 13C10 + 13C11 + 13C12 + 13C13

Using the formula for combinations, we can calculate each term:

13C0 = 1 (there is only one way to select 0 objects out of 13)
13C1 = 13 (there are 13 ways to select 1 object out of 13)
13C2 = 78
13C3 = 286
13C4 = 715
13C5 = 1287
13C6 = 1716
13C7 = 1716
13C8 = 1287
13C9 = 715
13C10 = 286
13C11 = 78
13C12 = 13
13C13 = 1

Therefore, the total number of possible combinations is:

1 + 13 + 78 + 286 + 715 + 1287 + 1716 + 1716 + 1287 + 715 + 286 + 78 + 13 + 1 = 8192

So, there are 8192 possible combinations with thirteen buttons. This can also be verified by using a combination calculator, which can generate all possible combinations for a given set of objects.

In conclusion, as a scientist, I would use principles of combinatorics to determine the number of possible combinations with thirteen buttons, taking