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There are ten lamps in the hall. Each lamp can be switched on and off independently. In how many ways can the hall be illuminated? (The hall is illuminated when at least one of the lamps is on)
(A) 10
(B) 100
(C) 128
(D) 1023
(E) 1024
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(A) 10
(B) 100
(C) 128
(D) 1023
(E) 1024
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Bunuel
Solution:
With each lamp, there are two options - ON or OFF
So, with 10 lamps, there are \(2^{10}=1024\) possible combinations of ON and OFF
In these 1024 different outcomes, there is one outcome where all the 10 lamps are OFF. We have to eliminate that
Thus total number of ways = 1024 - 1 = 1023
Hence the right answer is Option D
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