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Nested committee pairs

CombinatoricsHard

From a pool of $5$ distinct people, an ordered pair of committees $(A, B)$ is formed, where every member of $A$ is also in $B$ . Either committee may be empty, and $A$ may equal $B$ . How many ordered pairs exist?

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Hint

Classify each person independently by where they end up.

Solution

Each person has $3$ possible states: in neither committee, in $B$ only, or in both $A$ and $B$ .
The state "in $A$ only" is forbidden because $A \subseteq B$ .
The choices for different people are independent.
So the total is $3^{5}$ .
Therefore the answer is $243$ .

Answer: $243$

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