Difference between revisions of "2017 AMC 10A Problems/Problem 8"
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==Solution 2== | ==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>. | 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>. | ||
+ | |||
+ | ==Solution 3== | ||
+ | We can focus on how many handshakes the 10 people get. | ||
+ | The <math>1st</math> person gets 29 handshakes. | ||
+ | <math>2nd</math> gets 28 | ||
+ | ...... | ||
+ | And the <math>10th</math> gets 20 handshakes. | ||
+ | |||
+ | We can write this as the sum of an arithmetic sequence. | ||
+ | |||
+ | <math>\frac{10(20+29)}{2}\implies 5(49)\implies 245.</math> | ||
+ | Therefore, the answer is <math>\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 17:39, 19 April 2017
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 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. .
Solution 3
We can focus on how many handshakes the 10 people get. The person gets 29 handshakes. gets 28 ...... And the gets 20 handshakes.
We can write this as the sum of an arithmetic sequence.
Therefore, the answer is $\boxed{\textbf{(B)}\245}$ (Error compiling LaTeX. ! Undefined control sequence.)
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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