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

Revision as of 12:13, 3 October 2019

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?

$[asy] size(3cm); pair A[]; for (int i=0; i<9; ++i) { A[i] = rotate(22.5+45*i)*(1,0); } filldraw(A[0]--A[1]--A[2]--A[3]--A[4]--A[5]--A[6]--A[7]--cycle,gray,black); for (int i=0; i<8; ++i) { dot(A[i]); } [/asy]$

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 (Complementary)

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 $\frac{8\cdot6-8}{8\cdot7}=\boxed{\textbf{(D) } \frac 57}$

2018 AMC 8 (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 AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID PID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID PID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID PID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID PID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID
+==Solution 1==
-STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID PID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID PID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID PID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID PID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID PID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID PID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID PID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID PID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID PID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID PID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID PID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID PID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID PID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID STUPID
+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>.
+Thus our answer is <math>\frac{8+8\cdot4}{8\cdot7}= \frac{1+4}{7}=\boxed{\textbf{(D) } \frac 57}</math>
+==Solution 2 (Complementary)==
+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\cdot7}=\boxed{\textbf{(D) } \frac 57}</math>
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Difference between revisions of "2018 AMC 8 Problems/Problem 23"

Revision as of 12:13, 3 October 2019

Solution 1

Solution 2 (Complementary)

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