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

m (Problem)
(Problem Type)
Line 32: Line 32:
{{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}}
[[Category:Introductory Combinatorics Problems]]

Revision as of 15:48, 18 June 2018


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 within the group?

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

Solution 1

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 $200 + 45 = \boxed{\textbf{(B)}\ 245}$.

Solution 2

We can also use complementary counting. First of all, $\dbinom{30}{2}=435$ handshakes or hugs occur. Then, if we can find the number of hugs, then we can subtract it from $435$ to find the handshakes. Hugs only happen between the 20 people who know each other, so there are $\dbinom{20}{2}=190$ hugs. $435-190= \boxed{\textbf{(B)}\ 245}$.

Solution 3

We can focus on how many handshakes the 10 people get.

The 1st person gets 29 handshakes.

2nd gets 28


And the 10th receives 20 handshakes.

We can write this as the sum of an arithmetic sequence.

$\frac{10(20+29)}{2}\implies 5(49)\implies 245.$ Therefore, the answer is $\boxed{\textbf{(B)}\ 245}$

See Also

2017 AMC 10A (ProblemsAnswer KeyResources)
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Invalid username
Login to AoPS