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# There are $10$ lamps in a hall. Each of them can be switched on independently. The no. of ways in which the hall can be illuminated is?

$\begin{array}{1 1} 10^2 \\ 1023 \\ 2^{10} \\ 10! \end{array}$

Since each light can be switched on independently,
there are two options for each bulb (on or off)
$\therefore\:$Total no. of ways the bulbs can be operated= $2^{10}$
This includes all the bulbs are kept in off so that the room is not illuminated.
$\therefore$ The no. of ways in which the hall is illuminated=$2^{10}-1=1023$