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. $102$
• B. $1023$
• C. $210$
• D. $10!$

Answer 1

The correct option is B

$1023$

Explanation for correct option

There are $10$ lamps in a hall.

If at least one lamp is turned on, the hall can be illuminated

For each lamp there are $2$ choices $-$ on $/$ off

Therefore total number of ways $={}^{10}C_1+{}^{10}C_2+...+{}^{10}C_{10}$

$$\begin{gathered} =2^{10}-1;\left[\because {}^{n}C_1+{}^{n}C_2+....+{}^{n}C_n=2^n-1\right] \\ =1024-1 \\ =1023 \end{gathered}$$

Hence, option $\left(B\right)$ i.e. $1023$ is correct