2017 AMC 12A Problems/Problem 5
Contents
Problem
At a gathering of people, there are people who all know each other and people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?
Solution - Basic
All of the handshakes will involve at least one person from the who know no one. Label these ten people , , , , , , , , , .
Person from the group of 10 will initiate a handshake with everyone else ( people). Person initiates handshakes plus the one already counted from person . Person initiates new handshakes plus the two we already counted. This continues until person initiates handshakes plus the nine we already counted from ... .
Solution
Let the group of people who all know each other be , and let the group of people who know no one be . Handshakes occur between each pair such that and , and between each pair of members in . Thus, the answer is
Solution - Complementary Counting
The number of handshakes will be equivalent to the difference between the number of total interactions and the number of hugs, which are and , respectively. Thus, the total amount of handshakes is
Solution #4
Each of the 10 people who do not know anybody will shake hands with all 20 of the people who do know each other. This means there will be at least handshakes. In addition, those 10 people will also shake hands with each other, giving us another handshakes. Therefore, there is a total of handshakes.
See Also
2017 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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 |
2017 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 4 |
Followed by Problem 6 |
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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