There are 10 lamps in a hall.

We know that,

nCr

$=\frac{n !}{r !(n-r) !}$

We also know that,

$\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{C}_{\mathrm{k}}^{\mathrm{n}}=2^{\mathrm{n}}-1$

According to the question,

Number of lamps in a hall =10

Given that,

One of the lamps can be switched on independently

Hence, the number of ways in which the hall can be illuminated is given by,

C110 + C210 + C310 + C410 + C510 + C610 + C710 + C810 + C910 + C1010

=210-1

=1024-1

=1023