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| ==Problem==
| | #redirect [[2025 AMC 12A Problems/Problem 6]] |
| Six chairs are arranged around a round table. Two students and two teachers randomly select four of the chairs to sit in. What is the probability that the two students will sit in two adjacent chairs and the two teachers will also sit in two adjacent chairs?
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| <imath>\textbf{(A) } \frac 16 \qquad \textbf{(B) } \frac 15 \qquad \textbf{(C) } \frac 29 \qquad \textbf{(D) } \frac 3{13} \qquad \textbf{(E) } \frac 14</imath>
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| ==Video Solution==
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| https://youtu.be/CCYoHk2Af34
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| ==Solution 1==
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| Since rotations do not matter, we fix one student's seat. The next student can sit in any of the <imath>5</imath> remaining seats, but only two of these are successful, so there is a <imath>\dfrac25</imath> probability for the two students to sit next to each other. Regardless of the students' seats, there are now <imath>4</imath> seats remaining for the teachers. There are <imath>\dbinom42=6</imath> ways for the two teachers to sit, and of these, <imath>3</imath> of them have the two teachers sitting adjacent to one another. Therefore, the probability that the two teachers sit next to each other is <imath>\dfrac12.</imath> Multiplying the two probabilities gives <imath>\dfrac25\cdot\dfrac12=\boxed{\dfrac15\quad\textbf{(B)}.}</imath>
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| ==Solution 2==
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| First, we count the number of desired outcomes (<imath>2</imath> students sit together, <imath>2</imath> teachers sit together). We'll start by treating the <imath>2</imath> students as a block and doing the same for the teachers. Hence, we are seating <imath>2</imath> blocks in round table with <imath>4</imath> seats. It doesn't matter where we sit the student block since we can just rotate the table so they're at the top. After the student block is seated, there are <imath>3</imath> open seats for the teacher block. Since both students and both teachers can switch seats in their blocks, there are <imath>3\cdot2^2=12</imath> desired outcomes.
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| For the total outcomes, we first place one of the students, say Jimmy. It doesn't matter where we place Jimmy, as we can always rotate the table so he's at the top. Then, there are <imath>5\cdot4\cdot3</imath> ways to place the other student and the two teachers. There are hence <imath>5\cdot4\cdot3=60</imath> total outcomes. The answer is <imath>\boxed{\textbf{(B) }\dfrac{1}{5}}.</imath>
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| ~Tacos_are_yummy_1
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| ~minor <imath>\LaTeX</imath> edits by i_am_not_suk_at_math (saharshdevaraju 14:29, 7 November 2025 (EST)saharshdevaraju)
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| ===Alternate (slightly quicker) ending===
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| Once finding that the denominator is a multiple of 5, and reasoning that there is no 5 being multiplied in the numerator, the only possible answer choice is <imath>\boxed{\text{(B) }\dfrac{1}{5}}</imath> because all the other choices imply the numerator was a multiple of 5, and got cancelled out.
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| ~vaishnav
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| ==Solution 3==
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| There are <imath>6</imath> ways to select the first pair of adjacent chairs and <imath>3</imath> ways to select the next. There are <imath>2!</imath> ways to permute the students and teachers amongst themselves and <imath>\binom{6}{4} \cdot (2!)^2</imath> total ways they can sit down, giving us <imath>\frac{6 \cdot 3 \cdot (2!)^2}{\binom{6}{4} \cdot (2!)^2} = \boxed{\text{(B) }\dfrac{1}{5}}.</imath>
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| ~[[User:grogg007|grogg007]]
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| ==Solution 4==
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| First, we count the total number of outcomes: there are <imath>6</imath> choices for student <imath>1</imath>, <imath>5</imath> for student <imath>2</imath>, <imath>4</imath> for teacher <imath>1</imath>, and <imath>3</imath> for teacher <imath>2</imath> to make a total of <imath>6 * 5* 4 * 3 = 360</imath> total outcomes.
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| Next, we count the desired ones. There are <imath>6</imath> choices for student <imath>1</imath>, <imath>2</imath> for student <imath>2</imath>, and <imath>3</imath> for the two teachers, however, since they are different and have order, there is <imath>6 * 2 * 3 * 2! = 72</imath> desired outcomes.
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| Therefore, our answer is <imath>\frac{72}{360} = \boxed{\frac{1}{5}}</imath>
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| ~iiiiiizh
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| ==Solution 5==
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| There are six seats for a student to choose, Now, the next student can sit to the left or right to first, giving a <imath>\frac{2}{5}</imath> probability for the two students to sit next to each other. As for the teachers, the first one has four seats to choose from, but two of them will be next to the students. As for this case you get <imath>\frac{1}{2}</imath>, and then multiply by <imath>\frac{1}{3}</imath>, which is the chance that the second teacher sits next to the first. Applying the same process for the other two seats gives <imath>{\frac{1}{2}}\cdot{\frac{2}{3}}=\frac{1}{3}</imath>. Adding the two probabilities gives <imath>\frac{1}{2}</imath>, and mulitplying by the <imath>\frac{2}{5}</imath> we got earlier from the students we get a final answer of <imath>\boxed{\frac{1}{5}}</imath>.
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| ~Avy11
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| ===Alternate Solution (fractions at the end)===
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| If we start by picking the students first, then there is clearly 6 ways for one student to pick a spot, and 2 ways for the next student to choose a spot next to the other student. Then there are 4 spots for a techer to sit, but here is where we start casework.
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| <imath>\textbf{Case 1: Teacher chooses spot next to student}</imath>
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| If the teacher chooses a spot next to the student, there are 2 ways for the first teacher to sit, then there is only <imath>\boxed{1}</imath> other way for the other teacher to sit next to the first teacher.
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| <imath>\textbf{Case 2: Teacher doesn't sit next to student}</imath>
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| If the teacher doesn't sit next to the student, there is two ways to seat the first teacher, then there are <imath>\boxed{2}</imath> ways for the other teacher to sit next to the first teacher.
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| <imath>\color{red}{\textbf{BE CAREFUL!:}}</imath> Since the people are not the same (haha I didn't know that different people is different), we didn't overcount.
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| Then in total, there is <imath>6 \cdot 2 \cdot 2 \cdot (2+1)</imath>, and then there are <imath>6 \cdot 5 \cdot 4 \cdot 3</imath> ways in total so
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| <imath>\frac{6 \cdot 2 \cdot 2 \cdot 3}{6 \cdot 5 \cdot 4 \cdot 3}</imath> which simplifies to <imath>\boxed{\frac{1}{5}}</imath>.
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| ~Alzwang
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| =Solution 6=
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| Draw a circle
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| Label 6 points along its perimeter
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| Choose one for a teacher.
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| (This is just to make it easier to visualize)
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| Find the probability of the other teacher sitting next to the original one.
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| Separate the probability calculations for the students for the remaining four seats.
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| For the seats next to the teachers, there is only one place for the second student to sit.
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| For the seats not next to a teacher, there are two possibilities for where to sit.
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| Add up the possibilities for students, then multiply that by the teacher's probability.
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| ~ Rames Jong
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| == Video Solution (Fast and Easy) ==
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| https://youtu.be/sLhVhTM9HmA?si=3CeX_qoB7_GHdbHs ~ Pi Academy
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| ==Video Solution by SpreadTheMathLove==
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| https://www.youtube.com/watch?v=dAeyV60Hu5c
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| ==Video Solution==
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| https://youtu.be/gWSZeCKrOfU
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| ~MK
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| ==Video Solution by Daily Dose of Math==
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| https://youtu.be/5Fjos1vBt0A
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| ~Thesmartgreekmathdude
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| ==See Also==
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| {{AMC10 box|year=2025|ab=A|num-b=13|num-a=15}}
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| * [[AMC 10]]
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| * [[AMC 10 Problems and Solutions]]
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| * [[Mathematics competitions]]
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| * [[Mathematics competition resources]]
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| {{MAA Notice}}
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