Problem 24 · AMC 8 Stretch
Core
Counting & Probability
and-process-multiplycomplementary-counting
A committee of at least 2 people is to be formed from 5 people. How many different committees are possible?
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Answer: 26 committees
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Hint 1 of 3
A committee is just a subset of the 5 people. Start from all subsets, \(2^5\).
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Hint 2 of 3
'At least 2' throws out the empty committee and every one-person committee. Count those forbidden ones.
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Hint 3 of 3
Subtract the empty committee (1) and the singletons (5 of them) from \(2^5\).
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Approach: Complementary counting — subtract the too-small committees
- A committee is a subset of the 5 people, so there are \(2^5 = 32\) subsets in all (each person in or out).
- 'At least 2' forbids the empty committee and the 1-person committees. Empty committee: 1. One-person committees: 5.
- Subtract: \(32 - 1 - 5 = 26\).
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