Difference between revisions of "2013 AMC 10A Problems/Problem 11"

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==Problem==
 
==Problem==
  
A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly 10 ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person planning committee be selected?
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==YEETING IS BANNED!!!==
  
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==HOW TO CONTINUE THE UNO REVERSE CARD!!!==
  
<math> \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 25 </math>
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==YEET SO MUCH THAT THEY HAVE TO KICK EVERYONE, OR ALLOW YEETING AGAIN!!!==
 
 
==Solution==
 
 
 
Let the number of students on the council be <math>x</math>.  To select a two-person committee, we can select a "first person" and a "second person." There are <math>x</math> choices to select a first person; subsequently, there are <math>x-1</math> choices for the second person. This gives a preliminary count of <math>x(x-1)</math> ways to choose a two-person committee. However, this accounts for the order of committees. To understand this, suppose that Alice and Bob are two students in the council. If we choose Alice and then Bob, that is the same as choosing Bob and then Alice; however in our counting, the latter and former arrangements would be considered the same. Therefore, we have to divide by <math>2</math> to account for overcounting. Thus, there are <math>\dfrac{x(x-1)} 2=20</math> ways to choose the two-person committee. Solving this equation, we find that <math>5</math> and <math>-4</math> are integer solutions. <math>-4</math> is a ridiculous situation, so there are <math>5</math> people on the student council. The sol is <math>\dbinom 5 3=10\implies \boxed{\textbf{A}}</math>.
 
  
 
==See Also==   
 
==See Also==   
{{AMC10 box|year=2013|ab=A|num-b=10|num-a=12}}
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==BEST WAYS TO YEET!!!==
{{MAA Notice}}
 

Revision as of 21:29, 10 November 2019

Problem

YEETING IS BANNED!!!

HOW TO CONTINUE THE UNO REVERSE CARD!!!

YEET SO MUCH THAT THEY HAVE TO KICK EVERYONE, OR ALLOW YEETING AGAIN!!!

See Also

BEST WAYS TO YEET!!!

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