2003 AMC 12A Problem 22

Objects

A

and

B

move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object

A

starts at

(0, 0)

and each of its steps is either right or up, both equally likely. Object

B

starts at

(5, 7)

and each of its steps is either left or down, both equally likely. Which of the following is closest to the probability that the objects meet?

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Solution

The objects are

12

steps apart, so they can only meet after each takes

6

steps, on the anti-diagonal

x+y=6.

Pairing

A

’s six-step path with

B

’s reversed six-step path matches meeting pairs one-to-one with the

\binom{12}{5}

monotone walks from

(0,0)

(5,7).

The probability is

\dfrac{\binom{12}{5}}{2^{12}}=\dfrac{792}{4096}\approx0.19,

which is closest to

0.20.

Thus, the correct answer is C.

Tagged: lattice paths · random walk · bijection