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

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

Answer 1

The correct option is C

$1023$

Compute the number of ways:

If more than one lamp is on then the room is illuminated.

A lamp has two choices on or off.

Thus, number of ways$=(2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2)-1$

We subtract $1$, because there is one case when all lights are off and this is unwanted condition.

$\Rightarrow$Number of ways $=2^{10}-1$

$\Rightarrow$Number of ways $=1024–1$

$\Rightarrow$Number of ways $=1023$

Hence option (C) is the correct option.