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2013 AMC 12B Problems/Problem 12: Difference between revisions

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==Problem==
==Problem 12==


Cities <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, and <math>E</math> are connected by roads <math>AB</math>, <math>AD</math>, <math>AE</math>, <math>BC</math>, <math>BD</math>, <math>CD</math>, and <math>DE</math>. How many different routes are there from <math>A</math> to <math>B</math> that use each road exactly once? (Such a route will necessarily visit some cities more than once.)
Cities <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, and <math>E</math> are connected by roads <math>\widetilde{AB}</math>, <math>\widetilde{AD}</math>, <math>\widetilde{AE}</math>, <math>\widetilde{BC}</math>, <math>\widetilde{BD}</math>, <math>\widetilde{CD}</math>, and <math>\widetilde{DE}</math>. How many different routes are there from <math>A</math> to <math>B</math> that use each road exactly once? (Such a route will necessarily visit some cities more than once.)
<asy>
<asy>
unitsize(10mm);
unitsize(10mm);
Line 17: Line 17:
label("$D$",D,N);
label("$D$",D,N);
label("$E$",E,W);
label("$E$",E,W);
draw(A--B--C--D--E--cycle);
guide squiggly(path g, real stepsize, real slope=45)
draw(A--D);
{
draw(B--D);</asy>
real len = arclength(g);
real step = len / round(len / stepsize);
guide squig;
for (real u = 0; u < len; u += step){
real a = arctime(g, u);
real b = arctime(g, u + step / 2);
pair p = point(g, a);
pair q = point(g, b);
pair np = unit( rotate(slope) * dir(g,a));
pair nq = unit( rotate(0 - slope) * dir(g,b));
squig = squig .. p{np} .. q{nq};
}
squig = squig .. point(g, length(g)){unit(rotate(slope)*dir(g,length(g)))};
return squig;
}
pen pp = defaultpen + 2.718;
draw(squiggly(A--B, 4.04, 30), pp);
draw(squiggly(A--D, 7.777, 20), pp);
draw(squiggly(A--E, 5.050, 15), pp);
draw(squiggly(B--C, 5.050, 15), pp);
draw(squiggly(B--D, 4.04, 20), pp);
draw(squiggly(C--D, 2.718, 20), pp);
draw(squiggly(D--E, 2.718, -60), pp);</asy>


<math>\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 18</math>
<math>\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 18</math>


==Solution==
==Solution==

Revision as of 16:04, 23 February 2013

Problem 12

Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, and $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a route will necessarily visit some cities more than once.) [asy] unitsize(10mm); defaultpen(linewidth(1.2pt)+fontsize(10pt)); dotfactor=4; pair A=(1,0), B=(4.24,0), C=(5.24,3.08), D=(2.62,4.98), E=(0,3.08); dot (A); dot (B); dot (C); dot (D); dot (E); label("$A$",A,S); label("$B$",B,SE); label("$C$",C,E); label("$D$",D,N); label("$E$",E,W); guide squiggly(path g, real stepsize, real slope=45) { real len = arclength(g); real step = len / round(len / stepsize); guide squig; for (real u = 0; u < len; u += step){ real a = arctime(g, u); real b = arctime(g, u + step / 2); pair p = point(g, a); pair q = point(g, b); pair np = unit( rotate(slope) * dir(g,a)); pair nq = unit( rotate(0 - slope) * dir(g,b)); squig = squig .. p{np} .. q{nq}; } squig = squig .. point(g, length(g)){unit(rotate(slope)*dir(g,length(g)))}; return squig; } pen pp = defaultpen + 2.718; draw(squiggly(A--B, 4.04, 30), pp); draw(squiggly(A--D, 7.777, 20), pp); draw(squiggly(A--E, 5.050, 15), pp); draw(squiggly(B--C, 5.050, 15), pp); draw(squiggly(B--D, 4.04, 20), pp); draw(squiggly(C--D, 2.718, 20), pp); draw(squiggly(D--E, 2.718, -60), pp);[/asy]

$\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 18$


Solution

Note that cities $C$ and $E$ can be removed when counting paths because if a path goes in to $C$ or $E$, there is only one possible path to take out of cities $C$ or $E$. So the diagram is as follows: [asy] unitsize(10mm); defaultpen(linewidth(1.2pt)+fontsize(10pt)); dotfactor=4; pair A=(1,0), B=(4.24,0), C=(5.24,3.08), D=(2.62,4.98), E=(0,3.08); dot (A); dot (B); dot (D); label("$A$",A,S); label("$B$",B,SE); label("$D$",D,N); draw(A--B..D..cycle); draw(A--D); draw(B--D);[/asy]

Now we proceed with casework. Remember that there are two ways to travel from $A$ to $D$, $D$ to $A$, $B$ to $D$ and $D$ to $B$.:

Case 1 $A \Rightarrow D$: From $D$, if the path returns to $A$, then the next path must go to $B\Rightarrow D \Rightarrow B$. There are $2 \cdot 1 \cdot 2 = 4$ possibilities of the path $ADABDB$. If the path goes to $B$ from $D$, then the path must continue with either $BDAB$ or $BADB$. There are $2 \cdot 2 \cdot 2 = 8$ possibilities. So, this case gives $4+8=12$ different possibilities.

Case 2 $A \Rightarrow B$: The path must continue with $BDADB$. There are $2 \cdot 2 = 4$ possibilities for this case.

Putting the two cases together gives $12+4 = \boxed{\textbf{(D)}16}$

See also

2013 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
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 12 Problems and Solutions
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