Counting Problem?

Question

A large room has 5 windows and each window is either open or closed. How many different arrangements of open and close windows are there?

The answer is 32 but I have no clue why because I though it would be 10. Thanks!

Answer 1

Simplify the problem and consider first what the answer would be if there was only one window. This one window could be either open or closed, so there are two possibilities. Now lets add another window. Now for each possible arrangment of the first window (open or closed) we have two possibilities for the second window (open or closed). Thus there there are 2 times 2 = 4 possibilities for two windows. Now add a third window. For each of the four arrangments of the first two windows, the third window can be either open or closed and so there are 4 times 2 = 8 possibilities. Continuing like this you can see that for each new window you add, you multiply the number of arrangements by 2. Thus for five windows you have:

2x2x2x2x2 = 32

Answer 2

Alright, I'll break it down for you.

This is similar to the problem of:
If there are 5 people in the room, how many possible combinations of handshakes can there be (a shakes b's hand, a shakes c's hand, etc)
So with windows, we will say, "Alright, so one possible arrangement is that all the windows are closed. Now, open the first one. There's the second arrangement. Now, go back to them closed. Open the 2nd one. That's 3 arrangements. Go back to the original closed position. Now the 3rd window. Then 4th, and 5th. So how many combos do we have already?
6. See how that works?
Now, what you do is you open window #1, and start from there this time. Since you've already counted that possibility, we will open window #2, while #1 is already open. There's another arrangement. Now do #3 while #1 is open. Yet another combo.

You see where I am going with this? I hope I have made at least SOME sense in trying to explain this.

Good day.

Answer 3

list all the possibilites

closed closed closed closed closed

there is only 1 way you can rearrange the five closed windows

next:
open closed closed closed closed

There are 5! / 4! = 5 ways you can rearange an opened window and 4 closed windows
.

next:
opened opened closed closed closed

there are 5! / (2! * 3!) = 10 ways you can rearange two opened windows and three closed windows

next:
opened opened opened closed closed

there are 5! / (3! * 2!) = 10 you can rearange the three opened windows and 2 closed windows.


next:
opened opened openedn opened closed

there are 5! / 4! = 5 ways you can rearange four opened windows and one closed windown


next:
opened opened opened opened opened

there is only 1 way you can rearrange five opened windows


the total is 1 + 5 + 10 + 10 + 5 + 1 = 32

EDIT:

WOW!!!!! Dr. D, your method of solving this problem is like BOOM and you're done. Now, i learn something new. Thanks, Dr. D

Answer 4

Each window has 2 possibilities: open or closed.
Since there are 5 windows there are 2^5 = 32 possible combinations.

*EDIT*
Somebody's answer actually turns out to be the binomial expansion of (1+1)^5

Answer 5

its not ten b/c its counting like the 1st open the rest closed, all open, all closed, the 2nd and 3rd open the rest closed, like that. I think the easy way to find the answer is a rand button on a scientific calculator

Answer 6

I don't know how it would be 32 either? i got ten too