Question

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 $2^{10}$ -1.

• A. True
• B. False

Answer 1

The correct option is A True

Let us Assign the "on" state of 1 bulb be $1$

and the "off" state of 1 bulb be $0$

There is only 1 condition in which the room is not illuminated at all

which can be represented as 0000000000

other conditions like 0000000001,0111110000,....... are the ones in which the room will be illuminated

number of such conditions in which each bulb is either assigned $0$ or $1$ can be considered as:

There are 2 digits that are required to fill 10 places which can be expressed mathematically as $2\ast 2\ast 2\ast 2\ast 2\ast 2\ast 2\ast 2\ast 2\ast 2=2^{10}$

Subtracting one condition when every bulb is off

We get the total number $=2^{10}-1$