Difference between revisions of "2017 AMC 10A Problems/Problem 8"
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<math>\textbf{(A)}\ 240\qquad\textbf{(B)}\ 245\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 490</math> | <math>\textbf{(A)}\ 240\qquad\textbf{(B)}\ 245\qquad\textbf{(C)}\ 290\qquad\textbf{(D)}\ 480\qquad\textbf{(E)}\ 490</math> | ||
| − | ==Solution== | + | ==Solution 1== |
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 = 200</math>. 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>. | 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 = 200</math>. 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>. | ||
| + | |||
| + | ==Solution 2== | ||
| + | We can also use complementary counting. First of all, <math>\dbinom{30}{2}=435</math> handshakes or hugs occur. Then, if we can find the number of hugs, then we can subtract it from <math>435</math> to find the handshakes. Hugs only happen between the 20 people who know each other, so there are <math>\dbinom{20}{2}=190</math> hugs. <math>435-190= \boxed{\textbf{(B)}\ 245}</math>. | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2017|ab=A|num-b=7|num-a=9}} | {{AMC10 box|year=2017|ab=A|num-b=7|num-a=9}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 06:59, 9 February 2017
Contents
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?
Solution 1
Each one of the ten people has to shake hands with all the 
other people they don’t know. So 
. 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 
. Thus the answer is 
.
Solution 2
We can also use complementary counting. First of all, 
handshakes or hugs occur. Then, if we can find the number of hugs, then we can subtract it from 
to find the handshakes. Hugs only happen between the 20 people who know each other, so there are 
hugs. 
.
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 | ||
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