2018 AMC 8 Problems/Problem 23

LOOOOOOL

Solution

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 obviously 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 $8*4$ choices. The total amount of choices is ${8 \choose 3} = 8*7$ . Thus our answer is $\frac{8+8*4}{8*7}= \frac{1+4}{7}=\boxed{\textbf{(D) } \frac 57}$

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 $\frac{8*6-8}{8*7}=\boxed{\textbf{(D) } \frac 57}$

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

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2018 AMC 8 Problems/Problem 23

Solution

Solution 2 (Complementary)

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