2010 AMC 12A Problem 18

Problem 18 of 25IntermediateCounting & Probability

16

-step path is to go from

(-4,-4)

(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\le2,

-2\le y\le2

at each step?

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Solution

Every step increases

x+y

1,

which runs from

-8

8,

so each path passes through exactly one lattice point with

x+y=0.

To stay out of the open square, that point

(t,-t)

must have

|t|\ge2,

so it is one of

(\pm2,\mp2),(\pm3,\mp3),(\pm4,\mp4).

By symmetry consider the three points

(-4,4),(-3,3),(-2,2)

and double. The number of paths from

(-4,-4)

(-(4-j),4-j)

\binom{8}{j},

and the number continuing on to

(4,4)

is also

\binom{8}{j}.

Therefore the total is

\begin{gathered} 2\left(\binom80^2+\binom81^2+\binom82^2\right) \\ =2(1+64+784) \\ =1698. \end{gathered}

Thus, D is the correct answer.