2018 AMC 10B Problems/Problem 5
Contents
[hide]Problem
How many subsets of contain at least one prime number?
Solution 1 - Complementary Counting
We use complementary counting, or
.
There are a total of ways to create subsets (consider including or excluding each number) and there are a total of subsets only containing composite numbers (the composite numbers are ). Therefore, there are total ways to have at least one prime in a subset.
Solution 2 (Using Answer Choices)
Well, there are 4 composite numbers, and you can list them in a 1 number format, a 2 number, 3 number, and a 4 number format. Now, we can use combinations.
. Using the answer choices, the only multiple of 15 is
By: K6511
Solution 3
Subsets of include a single digit up to all eight numbers. Therefore, we must add the combinations of all possible subsets and subtract from each of the subsets formed by the composite numbers.
Hence:
By: pradyrajasai
Solution 4
Total subsets is Using complementary counting and finding the sets with composite numbers: only 4,6,8 and 9 are composite. Each one can be either in the set or out: = 16
-goldenn
Solution 5
Video Solution
~savannahsolver
See Also
2018 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
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All AMC 10 Problems and Solutions |
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