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Difference between revisions of "2010 AMC 12A Problems/Problem 18"

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<math>\textbf{(A)}\ 92 \qquad \textbf{(B)}\ 144 \qquad \textbf{(C)}\ 1568 \qquad \textbf{(D)}\ 1698 \qquad \textbf{(E)}\ 12,800</math>
 
<math>\textbf{(A)}\ 92 \qquad \textbf{(B)}\ 144 \qquad \textbf{(C)}\ 1568 \qquad \textbf{(D)}\ 1698 \qquad \textbf{(E)}\ 12,800</math>
  
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== Solution ==
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== Solution 1==
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=== Brute Force Solution ===
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Each path must go through either the second or the fourth quadrant.
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The number of ways to reach any point <math>(x,y)</math> on the grid is equal to the number of ways to reach <math>(x-1,y)</math> plus the number of ways to reach <math>(x,y-1)</math>. Using this recursion, we can draw the diagram and label each point with the number of ways to reach it and go up until we reach the end. Luckily, the figure is not so big that this is too time-consuming or difficult to do.
 
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For example:
 
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<cmath>\begin{align*}&(-4,-4) \Rightarrow 1\\
 
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&(-3,-4) \Rightarrow 1\\
 
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&(-2,-4) \Rightarrow 1\\
 
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&(-4,-3) \Rightarrow 1\\
 
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&(-4,-2) \Rightarrow 1\\
 
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&(-3,-3) \Rightarrow 2\\
 
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&(-2,-3) \Rightarrow 3\\
 
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&(-3,-2) \Rightarrow 3\\
 
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&(-2,-2) \Rightarrow 6\end{align*}</cmath>
 
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etc.
 
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We soon reach <math>(4,4) \Rightarrow 1698\ \boxed{\textbf{(D)}}</math>
 
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=== Combinatorial Solution 1 ===
 
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By symmetry we only need to count the paths that go through the second quadrant (<math>x<0</math>, <math>y>0</math>).
 
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For each of these paths, let <math>(a,2)</math> be the first point when it reaches <math>y=2</math>. Clearly <math>a\in\{-4,-3,-2\}</math> and the previous point on such path has to be <math>(a,1)</math>.
 
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Fix the value of <math>a</math>. There are <math>{a+9 \choose 5}</math> ways how the path can go from <math>(-4,-4)</math> to <math>(a,1)</math>, and then there are <math>{6-a \choose 2}</math> ways how the path can go from <math>(a,2)</math> to <math>(4,4)</math>.
 
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Hence for <math>a=-4</math> we get <math>{5 \choose 5}{10 \choose 2}=45</math> paths, for <math>a=-3</math> we get <math>{6 \choose 5}{9 \choose 2}=6\cdot 36=216</math> paths, and for <math>a=-2</math> we get <math>{7 \choose 5}{8 \choose 2}=21\cdot 28=588</math> paths. This gives us <math>45+216+588 = 849</math> paths through the second quadrant, hence the total number of paths is <math>2\cdot 849 = 1698</math>.
 
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=== Combinatorial Solution 2 ===
 
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Each path that goes through the second quadrant must pass through exactly one of the points <math>(-4,4)</math>, <math>(-3,3)</math>, and <math>(-2,2)</math>.
 
Each path that goes through the second quadrant must pass through exactly one of the points <math>(-4,4)</math>, <math>(-3,3)</math>, and <math>(-2,2)</math>.
  
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There is exactly <math>1</math> path of the first kind, <math>{8\choose 1}^2=64</math> paths of the second kind, and <math>{8\choose 2}^2=28^2=784</math> paths of the third type. The conclusion remains the same.
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There is <math>1</math> path of the first kind, <math>{8\choose 1}^2=64</math> paths of the second kind, and <math>{8\choose 2}^2=28^2=784</math> paths of the third type.  
 +
Each path that goes through the fourth quadrant must pass through exactly one of the points <math>(4,-4)</math>, <math>(3,-3)</math>, and <math>(2,-2)</math>.
 +
Again, there is <math>1</math> path of the first kind, <math>{8\choose 1}^2=64</math> paths of the second kind, and <math>{8\choose 2}^2=28^2=784</math> paths of the third type.  
  
 +
Hence the total number of paths is <math>2(1+64+784) = \boxed{1698}</math>.
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== Solution 2==
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We will use the concept of complimentary counting. If you go in the square you have to go out by these labeled points(click the link) https://imgur.com/VysX4P0 and go out through the borders because if you didnt, you would touch another point in one of those points in the set of points in the link(Call it <math>S</math>). There is symmetry about <math>y=x</math>, so we only have to consider <math>(1,-1), (1,0)</math>, and <math>(1,1)</math>. <math>(1,1)</math> can go on the boundary in two ways, so we can only consider one case and multiply it by two. For <math>(1,0)</math> and <math>(1,-1)</math> we can just multiply by two. So we count paths from <math>(-4,4)</math> to each of these points, and then multiply that by the number of ways to get from the point one unit right of that to <math>(4,4)</math>, and all in all, we get the answer is <math>\dbinom{16}{8}-2[\dbinom{8}{3}\dbinom{7}{2}+\dbinom{9}{4}\dbinom{6}{2}+\dbinom{10}{5}\dbinom{5}{2}]=1698</math>, which is answer choice <math>\textbf{(D)}</math>
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-vsamc
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Note: Sorry if this was rushed.
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2010|num-b=17|num-a=19|ab=A}}
 
{{AMC12 box|year=2010|num-b=17|num-a=19|ab=A}}
  
 
[[Category:Introductory Combinatorics Problems]]
 
[[Category:Introductory Combinatorics Problems]]
 +
{{MAA Notice}}

Revision as of 11:04, 11 March 2020

Problem

A 16-step path is to go from $(-4,-4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $-2 \le x \le 2$, $-2 \le y \le 2$ at each step?

$\textbf{(A)}\ 92 \qquad \textbf{(B)}\ 144 \qquad \textbf{(C)}\ 1568 \qquad \textbf{(D)}\ 1698 \qquad \textbf{(E)}\ 12,800$

Solution 1

Each path must go through either the second or the fourth quadrant. Each path that goes through the second quadrant must pass through exactly one of the points $(-4,4)$, $(-3,3)$, and $(-2,2)$.

There is $1$ path of the first kind, ${8\choose 1}^2=64$ paths of the second kind, and ${8\choose 2}^2=28^2=784$ paths of the third type. Each path that goes through the fourth quadrant must pass through exactly one of the points $(4,-4)$, $(3,-3)$, and $(2,-2)$. Again, there is $1$ path of the first kind, ${8\choose 1}^2=64$ paths of the second kind, and ${8\choose 2}^2=28^2=784$ paths of the third type.

Hence the total number of paths is $2(1+64+784) = \boxed{1698}$.

Solution 2

We will use the concept of complimentary counting. If you go in the square you have to go out by these labeled points(click the link) https://imgur.com/VysX4P0 and go out through the borders because if you didnt, you would touch another point in one of those points in the set of points in the link(Call it $S$). There is symmetry about $y=x$, so we only have to consider $(1,-1), (1,0)$, and $(1,1)$. $(1,1)$ can go on the boundary in two ways, so we can only consider one case and multiply it by two. For $(1,0)$ and $(1,-1)$ we can just multiply by two. So we count paths from $(-4,4)$ to each of these points, and then multiply that by the number of ways to get from the point one unit right of that to $(4,4)$, and all in all, we get the answer is $\dbinom{16}{8}-2[\dbinom{8}{3}\dbinom{7}{2}+\dbinom{9}{4}\dbinom{6}{2}+\dbinom{10}{5}\dbinom{5}{2}]=1698$, which is answer choice $\textbf{(D)}$ -vsamc Note: Sorry if this was rushed.

See also

2010 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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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