M10-15

Question

There are three lamps in a hall. Each lamp can be turned on or off independently. How many different ways can we light up the hall? (Note: The hall is considered lit if at least one lamp is on.)

A. 5
B. 6
C. 7
D. 8
E. 9

Bunuel · Answer

Official Solution:

There are three lamps in a hall. Each lamp can be turned on or off independently. How many different ways can we light up the hall? (Note: The hall is considered lit if at least one lamp is on.)

A. 5
B. 6
C. 7
D. 8
E. 9

Each lamp has two possible states: ON or OFF. Therefore, there are  2*2*2 = 8  potential combinations. However, since the case where all lamps are off is not considered a valid illumination, we subtract one, resulting in 7 valid ways to light up the hall.

Answer: C

rezaulnsu · Answer

If one lamp illuminates:3c1=3
If two lamps :3c2=3
If three lamps at a time=3c3=1
3+3+1=7

Posted from my mobile device

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Rod728 · Answer

Bunuel, how can I be sure that the order does not matter in this case? Given that the lamps are in different places, if Lamp 1 is ON and Lamps 2 and 3 are off, it is a different way of lighting the room than Lamp 2 ON and lamp 3 and 1 off.

Thanks

Thanks

bb · Answer

This is a good question.

I think you are reading it as in how many ways can the process of lighting be done by someone vs. in how many ways can the lamps be lit if someone enters the hall and sees it after the lights have been lit. Since the question is asking in how many ways can the hall be lit (and not in how many ways can the lights be lit), the language indicates that the question refers to the finished product (hall is lit and you enter it after the lighting process was done) and that is why the sequence of lighting does not matter, but I will let Bunuel be the final authority on this one

Rod728

Bunuel, how can I be sure that the order does not matter in this case? Given that the lamps are in different places, if Lamp 1 is ON and Lamps 2 and 3 are off, it is a different way of lighting the room than Lamp 2 ON and lamp 3 and 1 off.

Thanks

Bunuel · Answer

The order in which the lamps are lit does matter, and this is accounted for in the solution. The 8 combinations represent all possible ways to light the 3 lamps:

Lamp 1   
  Lamp 2   
  Lamp 3   
 
  Yes 
 Yes  
 Yes  
 
  Yes  
 Yes  
 No 
 
  Yes  
 No 
 Yes  
 
  Yes  
 No 
 No 
 
  No 
 Yes  
 Yes  
 
  No 
 Yes  
 No 
 
  No 
 No 
 Yes 
 
   No  
  No  
  No

The last combination, where all lamps are off, is not valid for lighting the hall, leaving 7 valid ways to light it.

Hope it's clear.

Rod728 · Answer

Dear Bunuel,

Thank you for your quick reply. Now I understand the solution better

MyNameisFritz · Answer

You can also just brute force this question.
O - lamp on
F - lamp off

Here are the combinations:

OOO
OOF
OFO
FOO
FFO
FOF
OFF

7 combinations total.

bumpbot · Answer

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Lamp 1	Lamp 2	Lamp 3
Yes	Yes	Yes
Yes	Yes	No
Yes	No	Yes
Yes	No	No
No	Yes	Yes
No	Yes	No
No	No	Yes
No	No	No