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 1 (Basic)
All of the handshakes will involve at least one person from the who knows no one. Label these ten people , , , , , , , , , .
Person from the group of 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 2
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 3 (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 people who do not know anybody will shake hands with all of the people who do know each other. This means there will be at least handshakes. In addition, those people will also shake hands with each other, giving us another handshakes. Therefore, there is a total of handshakes.
Solution 5
Every one of the people who know each other will shake hands with each of the people who know no one, so there are handshakes here. Each of the people will also shake hands with each other, so there will be handshakes for this case. In total, there are handshakes.
-Benedict T (countmath1)
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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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