2015 AMC 10A Problems/Problem 22

The following problem is from both the 2015 AMC 12A #17 and 2015 AMC 10A #22, so both problems redirect to this page.

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?

$\textbf{(A)}\dfrac{47}{256}\qquad\textbf{(B)}\dfrac{3}{16}\qquad\textbf{(C) }\dfrac{49}{256}\qquad\textbf{(D) }\dfrac{25}{128}\qquad\textbf{(E) }\dfrac{51}{256}$

Solution

Solution 1

We count how many valid arrangements there are and then divide by $2^8=256$ .

First, arbitrarily pick a person A and consider whether he is standing or sitting. If he is sitting, then the number of valid arrangements is the same as the number of valid arrangements of 7 people in a line. If he is 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 $a_n$ denote the number of ways to arrange $n$ people in a line such that no two adjacent people are standing. We determine a recurrence relation for $a_n,\ n\geq 2$ as follows. If the first person in the line is standing, then the next person must be sitting, and there are $a_{n-2}$ ways to arrange the rest. If the first person in the line is sitting, then there are $a_{n-1}$ ways to arrange the rest. Thus, $a_n=a_{n-1}+a_{n-2}$ and using the fact that $a_0=1$ and $a_1=2$ , we get that $a_n$ is the $(n+2)$ nd Fibonacci number $F_{n+2}$ . In particular, $a_5=F_7=13$ and $a_7=F_9=34$ .

So our desired count is $a_7+a_5=34+13=47$ , and the answer is $\boxed{\textbf{(A) } \frac{47}{256}}$ .

Solution 2

We will count how many valid standing arrangements there are (counting rotations as distinct), and divide by $2^8 = 256$ at the end. We casework on how many people are standing.

Case $1:$ $0$ people are standing. This yields $1$ arrangement.

Case $2:$ $1$ person is standing. This yields $8$ arrangements.

Case $3:$ $2$ people are standing. This yields $\dbinom{8}{2} - 8 = 20$ arrangements, because the two people cannot be next to each other.

Case $4:$ $4$ people are standing. Then the people must be arranged in stand-sit-stand-sit-stand-sit-stand-sit fashion, yielding $2$ possible arrangements.

More difficult is:

Case $5:$ $3$ people are standing. First, choose the location of the first person standing ( $8$ choices). Next, choose $2$ of the remaining people in the remaining $5$ legal seats to stand, amounting to $6$ arrangements considering that these two people cannot stand next to each other. However, we have to divide by $3,$ because there are $3$ ways to choose the first person given any three. This yields $\dfrac{8 \cdot 6}{3} = 16$ arrangements for Case $5.$

Alternate Case $5:$ Use complementary counting. Total number of ways to choose 3 people from 8 which is $\dbinom{8}{3}$ . Sub-case $1:$ three people are next to each other which is $\dbinom{8}{1}$ . Sub-case $2:$ two people are next to each other and the third person is not $\dbinom{8}{1}$ $\dbinom{4}{1}$ . This yields $\dbinom{8}{3} - \dbinom{8}{1} - \dbinom{8}{1} \dbinom{4}{1} = 16$

Summing gives $1 + 8 + 20 + 2 + 16 = 47,$ and so our probability is $\boxed{\textbf{(A) } \dfrac{47}{256}}$ .

Solution 3

We will count how many valid standing arrangements there are counting rotations as distinct and divide by $256$ at the end. Line up all $8$ 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 $2$ 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 $4$ standing, there are ${4 \choose 4}=1$ ways to place them. For $3,$ there are ${3+2 \choose 3}=10$ ways. etc. Summing, we get ${4 \choose 4}+{5 \choose 3}+{6 \choose 2}+{7 \choose 1}+{8 \choose 0}=1+10+15+7+1=34$ ways.

Now we consider that the far right person can be standing as well, so we have ${3 \choose 3}+{4 \choose 2}+{5 \choose 1}+{6 \choose 0}=1+6+5+1=13$ ways

Together we have $34+13=47$ , and so our probability is $\boxed{\textbf{(A) } \dfrac{47}{256}}$ .

Solution 4

We will count how many valid standing arrangements there are (counting rotations as distinct), and divide by $2^8 = 256$ 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 $7$ people. If they stand, we count the arrangements with $6$ instead because the person second from the left must sit. We notice that this is the Fibonacci sequence, where with $1$ person there are two ways and with $2$ people there are three ways. Carrying out the Fibonacci recursion until we get to $8$ people, we find there are $55$ 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 $4$ people to stand in a line, which is $8$ from our sequence. Therefore our probability is $\frac{55 - 8}{256} = \boxed{\textbf{(A) } \dfrac{47}{256}}$

Solution 5

We will count the number of valid arrangements and then divide by $2^8$ at the end. We proceed with casework on how many people are standing.

Case $1:$ $0$ people are standing. This yields $1$ arrangement.

Case $2:$ $1$ person is standing. This yields $8$ arrangements.

Case $3:$ $2$ people are standing. To do this, we imagine having 6 people with tails in a line first. Notate "tails" with $T$ . Thus, we have $TTTTTT$ . Now, we look to distribute the 2 $H$ 's into the 7 gaps made by the $T$ 's. We can do this in ${7 \choose 2}$ 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 ${7 \choose 2}-1=20$ arrangements.

Case $4:$ $3$ people are standing. Similarly, we imagine 5 $T$ 's. Thus, we have $TTTTT$ . We distribute 3 $H$ 's into the gaps, which can be done ${6 \choose 3}$ 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: ${4 \choose 1}=4$ ) Thus, we have ${6 \choose 3}-4=16$ arrangements.

Case $5:$ $4$ people are standing. This can clearly be done in 2 ways: $HTHTHTHT$ or $THTHTHTH$ . This yields $2$ arrangements.

Summing the cases, we get $1+8+20+16+2=47$ arrangements. Thus, the probability is $\boxed{\textbf{(A) } \dfrac{47}{256}}$

Video Solution (Alternative)

If you didn't quite follow this process, then you can look at this Ruzczyk video: https://www.youtube.com/watch?v=krlnSWWp0I0. I

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

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