Question

An ordered pair $(m,n)$ of nonnegative integers is called “simple” if adding $m+n$ in base $10$ requires no carrying. Find the number of simple ordered pairs that sum to $1492.$

Answer

Answer: 300

Concepts:digits place value multiplication principle

Difficulty rating: 1490

Small Hint:

Treat the four decimal places independently

Big Hint:

A target digit $d$ can be split into an ordered pair of digits in $d+1$ ways without carrying

Solution:

For a target digit $d,$ there are $d+1$ ordered pairs of nonnegative digits with sum $d.$ The four digits $1,$ $4,$ $9,$ $2$ therefore give $2,$ $5,$ $10,$ $3$ choices independently. The answer is $2\cdot5\cdot10\cdot3=300.$

1987 AIME Problem 1

Answer: 300

Difficulty rating: 1490

Small Hint:

Big Hint:

Solution:

Problem 1 in Other Years