Difference between revisions of "2017 AMC 10A Problems/Problem 8"

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==Solution==
 
==Solution==
Each one of the ten people has to shake hands with all the 20 other people they don’t know. So <math>10\times 20</math> = 200. From there you also have to calculate how many handshakes occurred between the people who don’t know each other. Each person out of the 10 has to shake hands with 9 other people. That’s 90, however you have to take into account the overlap. For example, there's 10 people so...
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Each one of the ten people has to shake hands with all the <math>20</math> other people they don’t know. So <math>10\cdot20</math> = 200. From there, we calculate how many handshakes occurred between the people who don’t know each other. This is simply counting how many ways to choose two people to shake hands, or <math>\binom{10}{2} = 45</math>. Thus the answer is <math>200 + 45 = \boxed{\textbf{(B)} 245}</math>.
 
 
Person 1 shakes hands with people- 2,3,4,5,6,7,8,9,10
 
And person 2 shakes hands with people- 1,3,4,5,6,7,8,9,10 and so on.
 
 
 
There's overlap. Person 1 shakes hands with person 2 and then person 2 shakes hands with person 1. So 90 needs to be divided by 2 so the overlap is taken into account. From there, add 200 + 45 to get the answer. (B) 245.
 

Revision as of 16:41, 8 February 2017

Problem

At a gathering of 30 people, there are 20 people who all know each other and 10 people who know no one. People who know each other a hug, and people who do not know each other shake hands. How many handshakes occur?

$\textbf{(A)}\ 240\qquad\textbf{(B)}\ 245\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 490$

Solution

Each one of the ten people has to shake hands with all the $20$ other people they don’t know. So $10\cdot20$ = 200. From there, we calculate how many handshakes occurred between the people who don’t know each other. This is simply counting how many ways to choose two people to shake hands, or $\binom{10}{2} = 45$. Thus the answer is $200 + 45 = \boxed{\textbf{(B)} 245}$.

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