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

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).

@@ Line 3: / Line 3: @@
 <math>\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad \textbf{(E) }10</math>
+==Video Solution (CREATIVE THINKING)==
+https://youtu.be/YCr5GFcmdcI
+~Education, the Study of Everything
 ==Video Solution==
-https://www.youtube.com/watch?v=rxrQLNxESW0
+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 <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{A}</math> is the correct answer.
+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.
 ==Solution 2==

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