2019 AIME I Problem 2

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

All

20 \cdot 19

ordered pairs

(J, B)

with

B \neq J

are equally likely

For each

J

there are

19 - J

values of

B

with

B \ge J + 2,

so the favorable count is

1 + 2 + \cdots + 18

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

\begin{aligned} &\sum_{J=1}^{18} (19 - J) \\ &= 18 + 17 + \cdots + 1 = 171. \end{aligned}

The probability is

\frac{171}{380} = \frac{9}{20},

so

m + n = 9 + 20 = 29.

Problem 2 in other years