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Difference between revisions of "2014 AMC 8 Problems/Problem 11"

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==Problem==
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Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?
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<math>\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad \textbf{(E) }10</math>
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==Video Solution (CREATIVE THINKING)==
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https://youtu.be/YCr5GFcmdcI
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~Education, the Study of Everything
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==Video Solution==
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https://www.youtube.com/watch?v=rxrQLNxESW0  ~David
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https://youtu.be/pWLm41JtiCw ~savannahsolver
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==Solution 1==
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We can apply complementary counting and count the paths that DO go through the blocked intersection, which is <math>\dbinom{2}{1}\dbinom{3}{1}=6</math>. There are a total of <math>\dbinom{5}{2}=10</math> paths, so there are <math>10-6=4</math> paths possible. <math>\boxed{(\text{A})4}</math> is the correct answer.
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==Solution 2==
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We can make a diagram of the roads available to Jack.
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<asy>
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size(50);
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defaultpen(linewidth(0.8));
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pair A=(0,2),B=origin,C=(2,0),D=(2,2),E=(2,1),F=(3,2),G=(3,1),H=(3,0);
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draw(A--B--H--F--A);
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draw(D--C);
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draw(E--G);
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</asy>
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Then, we can simply list the possible routes.
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<asy>
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size(50);
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defaultpen(linewidth(0.8));
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pair A=(0,2),B=origin,C=(2,0),D=(2,2),E=(2,1),F=(3,2),G=(3,1),H=(3,0);
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draw(B--H--F);
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draw(D--C,dotted);
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draw(E--G,dotted);
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draw(F--A--B,dotted);
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</asy>
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<asy>
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size(50);
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defaultpen(linewidth(0.8));
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pair A=(0,2),B=origin,C=(2,0),D=(2,2),E=(2,1),F=(3,2),G=(3,1),H=(3,0);
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draw(B--C--E--G--F);
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draw(B--A--F,dotted);
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draw(D--E,dotted);
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draw(C--H--G,dotted);
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</asy>
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<asy>
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size(50);
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defaultpen(linewidth(0.8));
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pair A=(0,2),B=origin,C=(2,0),D=(2,2),E=(2,1),F=(3,2),G=(3,1),H=(3,0);
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draw(B--C--D--F);
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draw(B--A--D,dotted);
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draw(E--G,dotted);
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draw(F--H--C,dotted);
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</asy>
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<asy>
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size(50);
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defaultpen(linewidth(0.8));
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pair A=(0,2),B=origin,C=(2,0),D=(2,2),E=(2,1),F=(3,2),G=(3,1),H=(3,0);
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draw(B--A--F);
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draw(B--H--F,dotted);
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draw(D--C,dotted);
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draw(E--G,dotted);
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</asy>
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There are 4 possible routes, so our answer is <math>\boxed{A}</math>.
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Note: This is not recommended in more complicated problems (e.g. Jill's house is 1000 blocks east and 400 blocks north of Jack's house).
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==See Also==
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{{AMC8 box|year=2014|num-b=10|num-a=12}}
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{{MAA Notice}}

Latest revision as of 20:47, 6 January 2024

Problem

Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?

$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad \textbf{(E) }10$


Video Solution (CREATIVE THINKING)

https://youtu.be/YCr5GFcmdcI

~Education, the Study of Everything



Video Solution

https://www.youtube.com/watch?v=rxrQLNxESW0 ~David

https://youtu.be/pWLm41JtiCw ~savannahsolver

Solution 1

We can apply complementary counting and count the paths that DO go through the blocked intersection, which is $\dbinom{2}{1}\dbinom{3}{1}=6$. There are a total of $\dbinom{5}{2}=10$ paths, so there are $10-6=4$ paths possible. $\boxed{(\text{A})4}$ is the correct answer.

Solution 2

We can make a diagram of the roads available to Jack. [asy] size(50); defaultpen(linewidth(0.8)); pair A=(0,2),B=origin,C=(2,0),D=(2,2),E=(2,1),F=(3,2),G=(3,1),H=(3,0); draw(A--B--H--F--A); draw(D--C); draw(E--G); [/asy] Then, we can simply list the possible routes. [asy] size(50); defaultpen(linewidth(0.8)); pair A=(0,2),B=origin,C=(2,0),D=(2,2),E=(2,1),F=(3,2),G=(3,1),H=(3,0); draw(B--H--F); draw(D--C,dotted); draw(E--G,dotted); draw(F--A--B,dotted); [/asy] [asy] size(50); defaultpen(linewidth(0.8)); pair A=(0,2),B=origin,C=(2,0),D=(2,2),E=(2,1),F=(3,2),G=(3,1),H=(3,0); draw(B--C--E--G--F); draw(B--A--F,dotted); draw(D--E,dotted); draw(C--H--G,dotted); [/asy] [asy] size(50); defaultpen(linewidth(0.8)); pair A=(0,2),B=origin,C=(2,0),D=(2,2),E=(2,1),F=(3,2),G=(3,1),H=(3,0); draw(B--C--D--F); draw(B--A--D,dotted); draw(E--G,dotted); draw(F--H--C,dotted); [/asy] [asy] size(50); defaultpen(linewidth(0.8)); pair A=(0,2),B=origin,C=(2,0),D=(2,2),E=(2,1),F=(3,2),G=(3,1),H=(3,0); draw(B--A--F); draw(B--H--F,dotted); draw(D--C,dotted); draw(E--G,dotted); [/asy] There are 4 possible routes, so our answer is $\boxed{A}$.


Note: This is not recommended in more complicated problems (e.g. Jill's house is 1000 blocks east and 400 blocks north of Jack's house).

See Also

2014 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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 AJHSME/AMC 8 Problems and Solutions

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

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