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Question
There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated.
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Solution
Total number of lamps = 10
The total number of ways in which hall can be illuminated is equal to the number of selection of one or more items out of n different items.
i.e. nC1 + nC2 + nC3 + nC4 + ... + nCn = 2n – 1
From Binomial expansion
We have nC0 + nC1 + nC2 + ... + nCn = 2n
So total number of ways = 10C1 + 10C2 + 10C3 + ... + 10C10
= 210 – 1
= 1024 – 1
= 1023
Hence, the required number of possible ways = 1023.
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