Question

Jenn randomly chooses a number $J$ from $1, 2, 3, \ldots, 19, 20.$ Bela then randomly chooses a number $B$ from $1, 2, 3, \ldots, 19, 20$ distinct from $J.$ The value of $B - J$ is at least $2$ with a probability that can be expressed in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Answer

Solution:

There are $20 \cdot 19 = 380$ equally likely ordered pairs $(J, B)$ with $B \neq J.$ The condition $B \ge J + 2$ allows $19 - J$ choices of $B$ for each $J \le 18,$ so the number of favorable pairs is $\sum_{J=1}^{18} (19 - J) = 18 + 17 + \cdots + 1 = 171.$

The probability is $\frac{171}{380} = \frac{9}{20},$ so $m + n = 9 + 20 = 29.$

2019 AIME I Problem 2

Solution:

Problem 2 in Other Years