Difference between revisions of "2018 AMC 8 Problems/Problem 23"

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Problem 23

From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?

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$\textbf{(A) } \frac{2}{7} \qquad \textbf{(B) } \frac{5}{42} \qquad \textbf{(C) } \frac{11}{14} \qquad \textbf{(D) } \frac{5}{7} \qquad \textbf{(E) } \frac{6}{7}$

Solutions

Solution 0

Choose side lengths $a,b,c$ for the triangle, starting from the shortest side, and going clockwise: $a+b+c=8$ , where $a\leq b, a \leq c$ . Options are: 1,1,6 ; 1,2,5 ; 1,3,4 ; 1,4,3 ; 1,5,2 ; 2,2,4 ; 2,3,3. $\boxed(5/7)$ of these have a side with length 1, which corresponds to an edge of the octagon.

Solution 1

We will use constructive counting to solve this. There are $2$ cases: Either all $3$ points are adjacent, or exactly $2$ points are adjacent.

If all $3$ points are adjacent, then we have $8$ choices. If we have exactly $2$ adjacent points, then we will have $8$ places to put the adjacent points and also $4$ places to put the remaining point, so we have $8\cdot4$ choices. The total amount of choices is ${8 \choose 3} = 8\cdot7$ .

Thus our answer is $\frac{8+8\cdot4}{8\cdot7}= \frac{1+4}{7}=\boxed{\textbf{(D) } \frac 57}$

Solution 2

We can decide $2$ adjacent points with $8$ choices. The remaining point will have $6$ choices. However, we have counted the case with $3$ adjacent points twice, so we need to subtract this case once. The case with the $3$ adjacent points has $8$ arrangements, so our answer is $\frac{8\cdot6-8}{{8 \choose 3 }}$ $=\frac{8\cdot6-8}{8 \cdot 7 \cdot 6 \div 6}\Longrightarrow\boxed{\textbf{(D) } \frac 57}$

Solution 3 (Stars and Bars)

Let $1$ point of the triangle be fixed at the top. Then, there are ${7 \choose 2} = 21$ ways to chose===Solution 1=== the other 2 points. There must be $3$ spaces in the points and $3$ points themselves. This leaves 2 extra points to be placed anywhere. By stars and bars, there are 3 triangle points (n) and $2$ extra points (k-1) distributed so by the stars and bars formula, ${n+k-1 \choose k-1}$ , there are ${4 \choose 2} = 6$ ways to arrange the bars and stars. Thus, the probability is $\frac{(21 - 6)}{21} = \boxed{\frac{5}{7}}$ .

Simple Complementary Counting

By rotational symmetry, choose an arbitrary point for one vertex. Then choose one of the 5 non-adjacent vertices, out of 7 possible. Sum the number of remaining non-adjacent vertices, for all 5 cases: 3+2+2+2+3=12, out of 6 possible for each. These are non-edge triangles, so the probability of edge triangles is $1-\frac{12}{7\cdot 6}= 5/7$

Video Solution by OmegaLearn

https://youtu.be/5UojVH4Cqqs?t=2678

~ pi_is_3.14

Video Solutions

https://www.youtube.com/watch?v=VNflxl7VpL0

https://youtu.be/YeYDixFXsvA

~savannahsolver

gg

@@ Line 18: / Line 18: @@
 ===Solution 0===
-Choose side lengths <math>a,b,c</math> for the triangle, starting from the shortest side, and going clockwise: <math>a+b+c=8</math>, where <math>a\leq b, </math>a \leq c<math>. Options are: 1,1,6 ; 1,2,5 ; 1,3,4 ; 1,4,3 ; 1,5,2 ; 2,2,4 ; 2,3,3. </math>\boxed(5/7)<math> of these have a side with length 1, which corresponds to an edge of the octagon.
+Choose side lengths <math>a,b,c</math> for the triangle, starting from the shortest side, and going clockwise: <math>a+b+c=8</math>, where <math>a\leq b, a \leq c</math>. Options are: 1,1,6 ; 1,2,5 ; 1,3,4 ; 1,4,3 ; 1,5,2 ; 2,2,4 ; 2,3,3. <math>\boxed(5/7)</math> of these have a side with length 1, which corresponds to an edge of the octagon.
 ===Solution 1===
-We will use constructive counting to solve this. There are </math>2<math> cases: Either all </math>3<math> points are adjacent, or exactly </math>2<math> points are adjacent.
+We will use constructive counting to solve this. There are <math>2</math> cases: Either all <math>3</math> points are adjacent, or exactly <math>2</math> points are adjacent.
-If all </math>3<math> points are adjacent, then we have </math>8<math> choices. If we have exactly </math>2<math> adjacent points, then we will have </math>8<math> places to put the adjacent points and also </math>4<math> places to put the remaining point, so we have </math>8\cdot4<math> choices. The total amount of choices is </math>{8 \choose 3} = 8\cdot7<math>.
+If all <math>3</math> points are adjacent, then we have <math>8</math> choices. If we have exactly <math>2</math> adjacent points, then we will have <math>8</math> places to put the adjacent points and also <math>4</math> places to put the remaining point, so we have <math>8\cdot4</math> choices. The total amount of choices is <math>{8 \choose 3} = 8\cdot7</math>.
-Thus our answer is </math>\frac{8+8\cdot4}{8\cdot7}= \frac{1+4}{7}=\boxed{\textbf{(D) } \frac 57}<math>
+Thus our answer is <math>\frac{8+8\cdot4}{8\cdot7}= \frac{1+4}{7}=\boxed{\textbf{(D) } \frac 57}</math>
 ===Solution 2 ===
-We can decide </math>2<math> adjacent points with </math>8<math> choices. The remaining point will have </math>6<math> choices. However, we have counted the case with </math>3<math> adjacent points twice, so we need to subtract this case once. The case with the </math>3<math> adjacent points has </math>8<math> arrangements, so our answer is </math>\frac{8\cdot6-8}{{8 \choose 3 }}<math></math>=\frac{8\cdot6-8}{8 \cdot 7 \cdot 6 \div 6}\Longrightarrow\boxed{\textbf{(D) } \frac 57}<math>
+We can decide <math>2</math> adjacent points with <math>8</math> choices. The remaining point will have <math>6</math> choices. However, we have counted the case with <math>3</math> adjacent points twice, so we need to subtract this case once. The case with the <math>3</math> adjacent points has <math>8</math> arrangements, so our answer is <math>\frac{8\cdot6-8}{{8 \choose 3 }}</math><math>=\frac{8\cdot6-8}{8 \cdot 7 \cdot 6 \div 6}\Longrightarrow\boxed{\textbf{(D) } \frac 57}</math>
 ===Solution 3 (Stars and Bars)===
-Let </math>1<math> point of the triangle be fixed at the top. Then, there are </math>{7 \choose 2} = 21<math> ways to chose===Solution 1=== the other 2 points. There must be </math>3<math> spaces in the points and </math>3<math> points themselves. This leaves 2 extra points to be placed anywhere. By stars and bars, there are 3 triangle points (n) and </math>2<math> extra points (k-1) distributed so by the stars and bars formula, </math>{n+k-1 \choose k-1}<math>, there are </math>{4 \choose 2} = 6<math> ways to arrange the bars and stars. Thus, the probability is </math>\frac{(21 - 6)}{21} = \boxed{\frac{5}{7}}<math>.
+Let <math>1</math> point of the triangle be fixed at the top. Then, there are <math>{7 \choose 2} = 21</math> ways to chose===Solution 1=== the other 2 points. There must be <math>3</math> spaces in the points and <math>3</math> points themselves. This leaves 2 extra points to be placed anywhere. By stars and bars, there are 3 triangle points (n) and <math>2</math> extra points (k-1) distributed so by the stars and bars formula, <math>{n+k-1 \choose k-1}</math>, there are <math>{4 \choose 2} = 6</math> ways to arrange the bars and stars. Thus, the probability is <math>\frac{(21 - 6)}{21} = \boxed{\frac{5}{7}}</math>.
 ===Simple Complementary Counting===
-By rotational symmetry, choose an arbitrary point for one vertex. Then choose one of the 5 non-adjacent vertices, out of 7 possible. Sum the number of remaining non-adjacent vertices, for all 5 cases: 3+2+2+2+3=12, out of 6 possible for each. These are non-edge triangles, so the probability of edge triangles is </math>1-\frac{12}{7\cdot 6}= 5/7$
+By rotational symmetry, choose an arbitrary point for one vertex. Then choose one of the 5 non-adjacent vertices, out of 7 possible. Sum the number of remaining non-adjacent vertices, for all 5 cases: 3+2+2+2+3=12, out of 6 possible for each. These are non-edge triangles, so the probability of edge triangles is <math>1-\frac{12}{7\cdot 6}= 5/7</math>
 == Video Solution by OmegaLearn ==

2018 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 22	Followed by Problem 24
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