Problem 8 · AMC 8 Stretch
Core
Counting & Probability
and-process-multiplyaccounting-for-all-possibilities
How many subsets does a set of 5 elements have? (A subset is any group you can form, including the empty group and the whole set.)
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Answer: 32 subsets
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Hint 1 of 3
Forming a subset is like the pizza problem: go through the elements one at a time and decide 'in' or 'out'.
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Hint 2 of 3
Each element gives 2 choices (in or out). How many elements are there?
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Hint 3 of 3
Multiply 2 by itself once for each element.
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Approach: AND process — an in/out choice for each element
- Forming a subset is a 5-step AND process: for each element, decide 'in' or 'out' (2 choices).
- The empty set (all 'out') and the full set (all 'in') both count.
- Multiply: \(2^5 = 32\). A set of 5 elements has 32 subsets.
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