"Eight switches are connected on a circuit so that if any one or more of the switches are closed, the light will go on. How many combinations of open and closed switches exist that will permit the light to go on?" (I suppose I should mention that this is under Permutations.)

After attempting to solve for 20 minutes, I checked the back of my textbook to find that the answer is 255 ways. I'm at a total loss for this discrete math and probability stuff and don't know how to get to that answer.

Question

"Eight switches are connected on a circuit so that if any one or more of the switches are closed, the light will go on. How many combinations of open and closed switches exist that will permit the light to go on?" (I suppose I should mention that this is under Permutations.)

After attempting to solve for 20 minutes, I checked the back of my textbook to find that the answer is 255 ways. I'm at a total loss for this discrete math and probability stuff and don't know how to get to that answer.

jim_thompson5910 · Answer

Think of each switch having two states: 0 or 1

You have 8 of these switches, so there are 2^8 = 256 different on/off (0 or 1) combinations. Only one of these 256 combos has all switches off and that is 00000000. 

The rest have at least one switch on. So you have 256-1 = 255 different ways to do this.

jim_thompson5910 · Answer

oops meant to say "on/off (1 or 0)" but you get the idea

mathslover · Answer

Was that so easy? I was trying using the formula .. Oh!

mathslover · Answer

You solved that very nicely @jim_thompson5910 - Awesome!

jim_thompson5910 · Answer

thanks mathslover

mathslover · Answer

But sir, will there be any case when all the switches are on ?

jim_thompson5910 · Answer

yes and that's represented by 11111111 (basically 8 ones)

mathslover · Answer

Yes sir, so if all the switches are on, how will the light go on? 

$\color{blue}{\text{Originally Posted by}}$ @tomhue
"Eight switches are connected on a circuit so that $\textbf{if any one or more of the switches are closed, the light will go on.}$
How many combinations of open and closed switches exist that will permit the light to go on?" 
$\color{blue}{\text{End of Quote}}$

mathslover · Answer

Have I misunderstood that point or something else?

jim_thompson5910 · Answer

well if all the switches are open (represented by 0), then we have 00000000

that's 1 case out of 256 total

the rest will have at least one switch closed

mathslover · Answer

Oh okay that's fine now. There was some confusion! :) 

Thanks again.

anonymous · Answer

Thanks a lot. I think I was just thinking too inside-the-box being given three formulas when it was referring to the previous section (on the fundamental counting principle). I guess I learned something there. :)

jim_thompson5910 · Answer

you're welcome

mathslover · Answer

Yeah Tom, I was thinking the same. *CLAPS* for Jim_Thompson ! Feel lucky that you're helped by him!