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A set 'S' has 5 elements. How many ways you can choose subset P and Q of S so that P∩Q is empty.

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We can choose P & Q in 243 different ways.
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Each of the 5 elements of set S will have 3 choices.
Either join subset P & don't join subset Q or
Either join subset Q  don't join subset P or
Neither join P nor join Q.
but it can join both P & Q since P intersection Q must be be an empty set.
Hence 3 choices for each of the 5 elements will gives us 3^5 = 243 subsets.
It is better to visualise such questions from the perspective of elements of the set that is try to figure out what choices each of these elements have.
Here is a similar question with nice explanation
http://aptitude.gateoverflow.in/280/empty-set-having-elements-and-are-two-subsets-such-that-subset
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