Question:
There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated.[Hint: Required number = 210 – 1].
Solution:
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