alleryStart learning →Why does a set of n distinct objects have exactly 2^n subsets?
AEach object is either included or left out
BEach object can be placed in n positions
CEach object is paired with one other object
DEach object can be repeated any number of times
Answer & Solution
Correct answer: A. Each object is either included or left out
1. Building a subset means going through the objects one at a time and deciding on each.
2. For every object there are exactly two decisions available: put it in or leave it out.
3. With n independent two-way decisions the Multiplication Principle gives 2 times 2 and so on, n times.
4. That product is 2^n, running from the empty subset up to the whole set.
5. Repetition plays no part, since each object appears at most once in a subset.
_Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 13 "Sequences, Probability and Counting Theory", section 13.5 Counting Principles_
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