2015 AMC 10B Problems/Problem 16
Contents
[hide]Problem
Al, Bill, and Cal will each randomly be assigned a whole number from to , inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's?
Video Solution
https://www.youtube.com/watch?v=vulB2z_PdRE&feature=youtu.be
Solution
We can solve this problem with a brute force approach.
- If Cal's number is :
- If Bill's number is , Al's can be any of .
- If Bill's number is , Al's can be any of .
- If Bill's number is , Al's can be .
- If Bill's number is , Al's can be .
- Otherwise, Al's number could not be a whole number multiple of Bill's.
- If Cal's number is :
- If Bill's number is , Al's can be .
- Otherwise, Al's number could not be a whole number multiple of Bill's while Bill's number is still a whole number multiple of Cal's.
- Otherwise, Bill's number must be greater than , i.e. Al's number could not be a whole number multiple of Bill's.
Clearly, there are exactly cases where Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal's. Since there are possible permutations of the numbers Al, Bill, and Cal were assigned, the probability that this is true is
-http://www.youtube.com/@OnDaTrainToMath
See Also
2015 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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All AMC 10 Problems and Solutions |
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