Let X is a set containing n elements. A subset P of set X is chosen at random. The set X is then reconstructed by replacing the elements of set Pand another set Q is chosen at random. Find number of ways to chosenP and Q such that P ∪ Q contains exactly r elements.
P ∪ Q has r elements. It means r elements out of n elements should be present in either P or in Q or both. r elements out of n elements can be selected in ways = nCr
Each of these elements has 3 choice
Rest (n – r) elements has 1 choice i.e. neither go in P nor in Q
⇒ Number of ways to select P and Q such
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