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?

$0.10$

$0.15$

$0.20$

$0.25$

$0.30$

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 $(0,0)$ 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.

2003 AMC 12A Problem 22

Solution:

Problem 22 in Other Years