2020 AMC 10B Problems/Problem 17
- The following problem is from both the 2020 AMC 10B #17 and 2020 AMC 12B #15, so both problems redirect to this page.
Contents
Problem
There are people standing equally spaced around a circle. Each person knows exactly of the other people: the people standing next to her or him, as well as the person directly across the circle. How many ways are there for the people to split up into pairs so that the members of each pair know each other?
Solution 1
Consider the people to be standing in a circle, where two people opposite each other form a diameter of the circle.
Let us use casework on the number of pairs that form a diameter of the circle.
Case 1: diameters
There are ways: either pairs with , pairs with , and so on or pairs with , pairs with , etc.
Case 2: diameter
There are possible diameters to draw (everyone else pairs with the person next to them).
Note that there cannot be diameters since there would be one person on either side that will not have a pair adjacent to them. The only scenario forced is when the two people on either side would be paired up across a diameter. Thus, a contradiction will arise.
Case 3: diameters
There are possible sets of diameters to draw. Notice we are technically choosing the number of ways to choose a pair of two diameters that are neighbors to each other. This means we can choose the first diameter in the pair, and have only two diameters to choose from for the second in the pair. This means we have possibilities for choosing 5 neighboring diameters. However, notice that there are duplicates, so we divide the possibilities by to get .
Note that there cannot be a case with diameters because then there would have to be diameters for the two remaining people as they have to be connected with a diameter. A contradiction arises.
Case 4: diameters
There is only way to do this.
Thus, in total there are possible ways. - Minor edits by Pearl2008
Solution 2
If each person knows exactly people, that means we form "-person groups". That is, all the people in the group knows every other person. We can rotate until the circle is filled. We want pairs of people, so the first pair has . The nd pair is just . We need to multiply these together since these are group. The rd pair would be . The th pair is . We multiply these together and get . The final group would be . So we add these up and we have possible ways.
(This is invalid. Imagine a circle of numbers. , , , , , , , , , . You might think that , , ,and is such a "-person group", but does not know or . Despite the answer being correct, this solution does not work. (Arcticturn or anybody, please correct me if I am wrong. I only THINK this solution is invalid and have provided my reasoning.))
~Arcticturn
-PerseverePlayer (Things in parenthesis)
Video Solution (HOW TO THINK CRITICALLY!!!)
~Education, the Study of Everything
Video Solution by TheBeautyOfMath
https://youtu.be/3BvJeZU3T-M?t=419
Video Solution by Sohil Rathi
https://youtu.be/0W3VmFp55cM?t=2796
See Also
2020 AMC 10B (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 10 Problems and Solutions |
2020 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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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