2003 AMC 12A Problem 22

Question

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?

0.10

0.15

0.20

0.25

0.30

They can meet only after each has taken

6

steps, along the line

x+y=6

The number of meeting path-pairs equals

\binom{12}{5}

out of

2^{12}

total pairs

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)

to

(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.

Problem 22 in other years