2018 AMC 12B Problems/Problem 5
Problem
How many subsets of contain at least one prime number?
Solution 1
Since an element of a subset is either in or out, the total number of subsets of the -element set is . However, since we are only concerned about the subsets with at least prime in it, we can use complementary counting to count the subsets without a prime and subtract that from the total. Because there are non-primes, there are subsets with at least prime.
Solution 2
We can construct our subset by choosing which primes are included and which composites are included. There are ways to select the primes (total subsets minus the empty set) and ways to select the composites. Thus, there are ways to choose a subset of the eight numbers.
Solution 3
We complement count. We know that we have subsets in all, including the empty subset. We subtract We have ways to choose a subset of the eight numbers that include a prime number. ~hh99754539
Video Solution by OmegaLearn
https://youtu.be/8WrdYLw9_ns?t=253
~ pi_is_3.14
See Also
2018 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 4 |
Followed by Problem 6 |
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All AMC 12 Problems and Solutions |
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