Q.A hall has 12 lamps and every lamp can be switched on independently. Find the number of ways of illuminating the hall.

When $n$ distinct objects are available and repetition IS allowed — meaning the same object may be used more than once across the $r$ chosen positions — the counting changes because using an object does not remove it from the available pool for the next position. Each of the $r$ positions can then independently be filled in all $n$ ways, regardless of what was chosen for the earlier positions. By the Multiplication Principle, the total number of such arrangements is $n\times n\times\cdots\times n$ ($r$ times), i.e. $n^r$. This situation arises whenever the 'objects' being placed are reusable categories rather than physical, single-use items — for example, forming a PIN code from 10 possible digits where the same digit can legitimately appear twice, assigning any of 3 flavours to each of several ice-cream scoops, or distributing distinguishable letters into post boxes where a box may receive mor …