Difference between revisions of "2010 AMC 12B Problems/Problem 16"
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Just like solution 1, we see that there is a <math>\frac{1}{3}</math> chance of <math>3|a</math> and <math>\frac{2}{9}</math> chance of <math>3|1+b+bc</math> | Just like solution 1, we see that there is a <math>\frac{1}{3}</math> chance of <math>3|a</math> and <math>\frac{2}{9}</math> chance of <math>3|1+b+bc</math> | ||
− | Now, we can just use PIE (Principals of Inclusion and Exclusion) to get our answer to be <math>\frac{1}{3}+\frac{2}{9}-\frac{1}{3}\cdot\frac{2}{9} = \boxed{\frac{13}{27}}</math> | + | Now, we can just use PIE (Principals of Inclusion and Exclusion) to get our answer to be <math>\frac{1}{3}+\frac{2}{9}-\frac{1}{3}\cdot\frac{2}{9} = \boxed{E) \frac{13}{27}}</math> |
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+ | -Conantwiz2023 | ||
== See also == | == See also == |
Revision as of 11:39, 4 February 2019
Problem 16
Positive integers , , and are randomly and independently selected with replacement from the set . What is the probability that is divisible by ?
Solution 1
We group this into groups of , because .
If , we are done. There is a probability of that that happens.
Otherwise, we have , which means that . So either or which will lead to the property being true. There are a chance for each bundle of cases to be true. Thus, the total for the cases is . But we have to multiply by because this only happens with a chance. So the total is actually .
The grand total is
Solution 2 (Minor change from Solution 1)
Just like solution 1, we see that there is a chance of and chance of
Now, we can just use PIE (Principals of Inclusion and Exclusion) to get our answer to be
-Conantwiz2023
See also
2010 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 15 |
Followed by Problem 17 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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