Difference between revisions of "2015 AMC 10A Problems/Problem 22"
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<math>\textbf{(A)}\dfrac{47}{256}\qquad\textbf{(B)}\dfrac{3}{16}\qquad\textbf{(C) }\dfrac{49}{256}\qquad\textbf{(D) }\dfrac{25}{128}\qquad\textbf{(E) }\dfrac{51}{256} </math> | <math>\textbf{(A)}\dfrac{47}{256}\qquad\textbf{(B)}\dfrac{3}{16}\qquad\textbf{(C) }\dfrac{49}{256}\qquad\textbf{(D) }\dfrac{25}{128}\qquad\textbf{(E) }\dfrac{51}{256} </math> | ||
− | == | + | ==Solutions== |
===Solution 1=== | ===Solution 1=== | ||
We count how many valid arrangements there are and then divide by <math>2^8=256</math>. | We count how many valid arrangements there are and then divide by <math>2^8=256</math>. | ||
− | First, arbitrarily pick a person A and consider whether | + | First, arbitrarily pick a person A and consider whether they are standing or sitting. |
− | If | + | If they are sitting, then the number of valid arrangements is the same as the number of valid arrangements of 7 people in a line. |
− | If | + | If they are standing, then the two people next to him must be sitting, so the number of valid arrangements is the same as the number of valid arrangements of 5 people in a line. |
Let <math>a_n</math> denote the number of ways to arrange <math>n</math> people in a line such that no two adjacent people are standing. We determine a recurrence relation for <math>a_n,\ n\geq 2</math> as follows. If the first person in the line is standing, then the next person must be sitting, and there are <math>a_{n-2}</math> ways to arrange the rest. If the first person in the line is sitting, then there are <math>a_{n-1}</math> ways to arrange the rest. Thus, <math>a_n=a_{n-1}+a_{n-2}</math> and using the fact that <math>a_0=1</math> and <math>a_1=2</math>, we get that <math>a_n</math> is the <math>(n+2)</math>nd Fibonacci number <math>F_{n+2}</math>. In particular, <math>a_5=F_7=13</math> and <math>a_7=F_9=34</math>. | Let <math>a_n</math> denote the number of ways to arrange <math>n</math> people in a line such that no two adjacent people are standing. We determine a recurrence relation for <math>a_n,\ n\geq 2</math> as follows. If the first person in the line is standing, then the next person must be sitting, and there are <math>a_{n-2}</math> ways to arrange the rest. If the first person in the line is sitting, then there are <math>a_{n-1}</math> ways to arrange the rest. Thus, <math>a_n=a_{n-1}+a_{n-2}</math> and using the fact that <math>a_0=1</math> and <math>a_1=2</math>, we get that <math>a_n</math> is the <math>(n+2)</math>nd Fibonacci number <math>F_{n+2}</math>. In particular, <math>a_5=F_7=13</math> and <math>a_7=F_9=34</math>. | ||
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https://www.youtube.com/watch?v=krlnSWWp0I0 | https://www.youtube.com/watch?v=krlnSWWp0I0 | ||
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== See Also == | == See Also == |
Revision as of 19:12, 8 December 2020
- The following problem is from both the 2015 AMC 12A #17 and 2015 AMC 10A #22, so both problems redirect to this page.
Contents
Problem
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
Solutions
Solution 1
We count how many valid arrangements there are and then divide by .
First, arbitrarily pick a person A and consider whether they are standing or sitting. If they are sitting, then the number of valid arrangements is the same as the number of valid arrangements of 7 people in a line. If they are standing, then the two people next to him must be sitting, so the number of valid arrangements is the same as the number of valid arrangements of 5 people in a line.
Let denote the number of ways to arrange people in a line such that no two adjacent people are standing. We determine a recurrence relation for as follows. If the first person in the line is standing, then the next person must be sitting, and there are ways to arrange the rest. If the first person in the line is sitting, then there are ways to arrange the rest. Thus, and using the fact that and , we get that is the nd Fibonacci number . In particular, and .
So our desired count is , and the answer is .
Solution 2
We will count how many valid standing arrangements there are (counting rotations as distinct), and divide by at the end. We casework on how many people are standing.
Case people are standing. This yields arrangement.
Case person is standing. This yields arrangements.
Case people are standing. This yields arrangements, because the two people cannot be next to each other.
Case people are standing. Then the people must be arranged in stand-sit-stand-sit-stand-sit-stand-sit fashion, yielding possible arrangements.
More difficult is:
Case people are standing. First, choose the location of the first person standing ( choices). Next, choose of the remaining people in the remaining legal seats to stand, amounting to arrangements considering that these two people cannot stand next to each other. However, we have to divide by because there are ways to choose the first person given any three. This yields arrangements for Case
Alternate Case Use complementary counting. Total number of ways to choose 3 people from 8 which is . Sub-case three people are next to each other which is . Sub-case two people are next to each other and the third person is not . This yields
Summing gives and so our probability is .
Solution 3
We will count how many valid standing arrangements there are counting rotations as distinct and divide by at the end. Line up all people linearly. In order for no two people standing to be adjacent, we will place a sitting person to the right of each standing person. In effect, each standing person requires spaces and the standing people are separated by sitting people. We just need to determine the number of combinations of pairs and singles and the problem becomes very similar to pirates and gold aka stars and bars aka sticks and stones aka balls and urns.
If there are standing, there are ways to place them. For there are ways. etc. Summing, we get ways.
Now we consider that the far right person can be standing as well, so we have ways
Together we have , and so our probability is .
Solution 4
We will count how many valid standing arrangements there are (counting rotations as distinct), and divide by at the end. If we suppose for the moment that the people are in a line, and decide from left to right whether they sit or stand. If the leftmost person sits, we have the same number of arrangements as if there were only people. If they stand, we count the arrangements with instead because the person second from the left must sit. We notice that this is the Fibonacci sequence, where with person there are two ways and with people there are three ways. Carrying out the Fibonacci recursion until we get to people, we find there are standing arrangements. Some of these were illegal however, since both the first and last people stood. In these cases, both the leftmost and rightmost two people are fixed, leaving us to subtract the number of ways for people to stand in a line, which is from our sequence. Therefore our probability is
Solution 5
We will count the number of valid arrangements and then divide by at the end. We proceed with casework on how many people are standing.
Case people are standing. This yields arrangement.
Case person is standing. This yields arrangements.
Case people are standing. To do this, we imagine having 6 people with tails in a line first. Notate "tails" with . Thus, we have . Now, we look to distribute the 2 's into the 7 gaps made by the 's. We can do this in ways. However, note one way does not work, because we have two H's at the end, and the problem states we have a table, not a line. So, we have arrangements.
Case people are standing. Similarly, we imagine 5 's. Thus, we have . We distribute 3 's into the gaps, which can be done ways. However, 4 arrangements will not work. (See this by putting the H's at the ends, and then choosing one of the remaining 4 gaps: ) Thus, we have arrangements.
Case people are standing. This can clearly be done in 2 ways: or . This yields arrangements.
Summing the cases, we get arrangements. Thus, the probability is
Video Solution by Richard Rusczyk
https://www.youtube.com/watch?v=krlnSWWp0I0
See Also
2015 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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 10 Problems and Solutions |
2015 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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.