How can permutations help with logic gates and circuit design?

[BOAS]

Hello,

today in class we started a topic on permutations and combinations and I have come across a way in which it could be of use to me whilst working with 'logictutor' (a premade circuit board used to teach simple logic circuits).

We have an experiment tomorrow where we will investigate different logic gates and build a decade counter and whilst my question isn't about that, I think it would be nice to show the permutations of inputs.

Homework Statement

There are four switches on the board that control the input to a seven segment display and I need to fill out a table that shows the position of each switch and what the display will show.

Homework Equations

The Attempt at a Solution

There are 4! permutations possible for 4 items but in my situation each item can be either on or off.

I hope this doesn't translate to their being 8! permutations otherwise I'm going to be up all night on excel making tables.

So how do I model this problem to show the number of permutations? I suppose I would like to know because as truth tables get bigger and bigger it could be a quick check to show I have every possibility.

Thanks! (I hope my question is clear)

 

[haruspex]

No factorials arise in this context. You are not shuffling the order of things or selecting fixed size subsets.
If you wanted to list the combinations in terms of how many switches are on then you would see some factorials, but that's not an interesting way to list them here.
You have N distinct things, each of which can be in any of R states. How many possible combinations of states?
If you're not sure, start with N=1 and work up.

 

[BOAS]

No factorials arise in this context. You are not shuffling the order of things or selecting fixed size subsets.
If you wanted to list the combinations in terms of how many switches are on then you would see some factorials, but that's not an interesting way to list them here.
You have N distinct things, each of which can be in either of R states. How many possible combinations of states?
If you're not sure, start with N=1 and work up.

One switch has 2 different combinations of states.

Two switches have four different combinations of states.

Three switches have 8 different combinations of states.

Looking at this, the general rule appears to be number of combinations = r^N

For my four switch scenario there should be 16 combinations - Which is precisely how many I could come up with :)

Thank you!

 

[phinds]

Yep, now you've got it.