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

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==Problem==
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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?
 
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?
  
  
 
<math> \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 25 </math>
 
<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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==Solution==
  
 
Let the number of students on the council be <math>x</math>.  We know that there are <math>\dbinom{x}{2}</math> ways to choose a two person team.  This gives that <math>x(x-1) = 20</math>, which has a positive integer solution of <math>5</math>.
 
Let the number of students on the council be <math>x</math>.  We know that there are <math>\dbinom{x}{2}</math> ways to choose a two person team.  This gives that <math>x(x-1) = 20</math>, which has a positive integer solution of <math>5</math>.
  
 
If there are <math>5</math> people on the welcoming committee, then there are <math>\dbinom{5}{3} = 10</math> ways to choose a three-person committee, <math>\textbf{(A)}</math>
 
If there are <math>5</math> people on the welcoming committee, then there are <math>\dbinom{5}{3} = 10</math> ways to choose a three-person committee, <math>\textbf{(A)}</math>

Revision as of 19:25, 7 February 2013

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?


$\textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 25$

Solution

Let the number of students on the council be $x$. We know that there are $\dbinom{x}{2}$ ways to choose a two person team. This gives that $x(x-1) = 20$, which has a positive integer solution of $5$.

If there are $5$ people on the welcoming committee, then there are $\dbinom{5}{3} = 10$ ways to choose a three-person committee, $\textbf{(A)}$