(B) Each lamp has $2$ possibilities: it can be either switched on or switched off.Since there are $10$ lamps,the total number of ways to switch them i

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(B) Each lamp has $2$ possibilities: it can be either switched on or switched off.Since there are $10$ lamps,the total number of ways to switch them is $2 \times 2 \times \dots \times 2$ ($10$ times),which equals $2^{10} = 1024$.However,the hall is illuminated only if at least one lamp is switched on.Therefore,we must exclude the case where all lamps are switched off.The number of ways to illuminate the hall is $2^{10} - 1 = 1024 - 1 = 1023$.

Class 11 Mathematics Permutation and Combination Definition of permutation, Number of permutations with or without repetition, Conditional permutations

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There are $10$ lamps in a hall. Each one of them can be switched on independently. The number of ways in which the hall can be illuminated is

A
$10^2$
B
$1023$
C
$2^{10}$
D
$10!$

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Permutation and Combination

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Definition of permutation, Number of permutations with or without repetition, Conditional permutations

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MediumAP EAMCET 2021

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There are $10$ lamps in a hall. Each one of them can be switched on independently. The number of ways in which the hall can be illuminated is

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