Difference between revisions of "2021 AMC 12B Problems/Problem 23"
Darren yao (talk | contribs) (→Solution) |
|||
Line 16: | Line 16: | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
The answer is <math>6+49=\boxed{\textbf{(A) }55}.</math> | The answer is <math>6+49=\boxed{\textbf{(A) }55}.</math> | ||
+ | |||
+ | ==Solution 2== | ||
+ | As in solution 1, note that "evenly spaced" means the bins are in arithmetic sequence. We let the first bin be <math>a</math> and the common difference be <math>d</math>. Further note that each <math>(a, d)</math> pair uniquely determines a set of 3 bins. | ||
+ | |||
+ | We have <math>1 \leq a \leq \infty</math> because the leftmost bin in the sequence can be any bin, and <math>1 \leq d \leq \infty</math>, because the bins must be distinct. | ||
+ | |||
+ | This gives us the following sum for the probability: | ||
+ | |||
+ | <cmath> 6 \sum_{a=1}^{\infty} \sum_{d=1}^{\infty} 2^{-3a-3d} </cmath> | ||
+ | |||
+ | <cmath> = 6 \sum_{a=1}^{\infty} \sum_{d=1}^{\infty} 2^{-3a} \cdot 2^{-3d} </cmath> | ||
+ | |||
+ | <cmath> = 6 \left( \sum_{a=1}^{\infty} 2^{-3a} \right) \left( \sum_{d=1}^{\infty} 2^{-3d} \right) </cmath> | ||
+ | |||
+ | <cmath> = 6 \left( \sum_{a=1}^{\infty} 8^{-a} \right) \left( \sum_{d=1}^{\infty} 8^{-d} \right) </cmath> | ||
+ | |||
+ | <cmath> = 6 \left( \frac{1}{7} \right) \left( \frac{1}{7} \right) </cmath> | ||
+ | |||
+ | <cmath> = \frac{6}{49} </cmath> | ||
+ | |||
+ | <cmath> \to 55 \text{ (A) }. </cmath> | ||
+ | |||
+ | -Darren Yao | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2021|ab=B|num-b=22|num-a=24}} | {{AMC12 box|year=2021|ab=B|num-b=22|num-a=24}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 15:17, 12 February 2021
Contents
[hide]Problem
Three balls are randomly and independantly tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin is for More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is where and are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins and ) What is
Solution
"Evenly spaced" just means the bins form an arithmetic sequence.
Suppose the middle bin in the sequence is . There are different possibilities for the first bin, and these two bins uniquely determine the final bin. Now, the probability that these bins are chosen is , so the probability is the middle bin is . Then, we want the sum The answer is
Solution 2
As in solution 1, note that "evenly spaced" means the bins are in arithmetic sequence. We let the first bin be and the common difference be . Further note that each pair uniquely determines a set of 3 bins.
We have because the leftmost bin in the sequence can be any bin, and , because the bins must be distinct.
This gives us the following sum for the probability:
-Darren Yao
See Also
2021 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.