Difference between revisions of "1978 AHSME Problems/Problem 16"
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==Problem== | ==Problem== | ||
In a room containing <math>N</math> people, <math>N > 3</math>, at least one person has not shaken hands with everyone else in the room. | In a room containing <math>N</math> people, <math>N > 3</math>, at least one person has not shaken hands with everyone else in the room. | ||
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\textbf{(E) }\text{none of these} </math> | \textbf{(E) }\text{none of these} </math> | ||
| − | + | ==Solution== | |
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| + | We assume that one person hasn't shaken hands with the other N people, meaning that they only had shaken hands with N-1 people. However, this doesn't make sense! There will be 1 person ( from the N-1) people who hasn't shaken hands with the first person, meaning that they also only had shaken hands with N-1 people. Therefore, there is a minimum of 2 people in the room that haven't shaken hands with one person. Therefore, the maximum is N-2 people, so the answer is <math>E</math> | ||
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| + | ~Brackie 1331 | ||
Latest revision as of 02:39, 1 June 2024
Problem
In a room containing 
people, 
, at least one person has not shaken hands with everyone else in the room.
What is the maximum number of people in the room that could have shaken hands with everyone else?
Solution
We assume that one person hasn't shaken hands with the other N people, meaning that they only had shaken hands with N-1 people. However, this doesn't make sense! There will be 1 person ( from the N-1) people who hasn't shaken hands with the first person, meaning that they also only had shaken hands with N-1 people. Therefore, there is a minimum of 2 people in the room that haven't shaken hands with one person. Therefore, the maximum is N-2 people, so the answer is 
~Brackie 1331
