2017 AMC 10B Problems/Problem 13

Problem

There are $20$ students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are $10$ students taking yoga, $13$ taking bridge, and $9$ taking painting. There are $9$ students taking at least two classes. How many students are taking all three classes?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

Solution 1

By PIE (Property of Inclusion/Exclusion), we have

$|A_1 \cup A_2 \cup A_3| = \sum |A_i| - \sum |A_i \cap A_j| + |A_1 \cap A_2 \cap A_3|.$ Number of people in at least two sets is $\sum |A_i \cap A_j| - 2|A_1 \cap A_2 \cap A_3| = 9.$ So, $20 = (10 + 13 + 9) - (9 + 2x) + x,$ which gives $x = \boxed{\textbf{(C) } 3}.$

Solution 2 (Subtraction)

The total number of classes taken among the 20 students is $10 + 13 + 9 = 32$ . Each student is taking at least one class so let's subtract the $20$ classes ( $1$ per each of the $20$ students) from $32$ classes to get $12$ . $12$ classes is the total number of extra classes taken by the students who take $2$ or $3$ classes. Since we know that there are $9$ students taking at least $2$ classes, there must be $12 - 9 = \boxed{\textbf{(C) } 3}$ students that are taking all $3$ classes.

Solution 3 (Algebra)

Total class count is 32. Assume there are $a$ students taking one class, $b$ students taking two classes, ad $c$ students taking three classes. Because there are $20$ students total, $a+b+c = 20$ . Because each student taking two classes is counted twice, and each student taking three classes is counted thrice in the total class count, $a+2b+3c = 10+13+9 = 32$ . There are $9$ students taking two or three classes, so $b+c = 9$ . Solving this system of equations gives us $c=\boxed{\textbf{(C) 3}}$ .

Solution 4 ( A Combination of the Principle of Inclusion/Exclusion and Algebra)

Let us assign the following variables: $a$ which designates the number of people taking exactly Bridge and Yoga. $b$ which designates the number of people taking exactly Bridge and Painting.

\documentclass{article}

\usepackage{tikz} \usetikzlibrary{shapes,backgrounds} \begin{document}

\def\firstcircle{(0,0) circle (1.5cm)} \def\secondcircle{(60:2cm) circle (1.5cm)} \def\thirdcircle{(0:2cm) circle (1.5cm)} \begin{tikzpicture} \begin{scope}[shift={(6cm,0cm)}]

       \begin{scope}[even odd rule]% first circle without the second
           \clip \thirdcircle (-3, -3) rectangle (3,3);
           \clip \secondcircle (-3, -3) rectangle (3,3);
       \fill[yellow] \firstcircle;

       \end{scope}
       \draw \firstcircle node {};
       \draw \secondcircle node {};
       \draw \thirdcircle node {};
   \end{scope}
 \end{tikzpicture}

2017 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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