1987 AIME Problems/Problem 1

Problem

An ordered pair $(m,n)$ of non-negative integers is called "simple" if the addition $m+n$ in base $10$ requires no carrying. Find the number of simple ordered pairs of non-negative integers that sum to $1492$ .

Solution

Since no carrying over is allowed, the range of possible values of any digit of $m$ is from $0$ to the respective digit in $1492$ (the values of $n$ are then fixed). Thus, the number of ordered pairs will be $(1 + 1)(4 + 1)(9 + 1)(2 + 1) = 2\cdot 5\cdot 10\cdot 3 = \boxed{300}$ .

If you do not understand the above solution, consider this. For every positive integer $m$ , there is only one whole number $n$ that you can add to it to obtain the required sum. Also, the total number of non-negative integers that are smaller than or equal to an integer $a$ is $(a + 1)$ because there are $(a - 1)$ positive integers that are less than it, and $0$ , in addition to itself.

1987 AIME (Problems • Answer Key • Resources)
Preceded by First Question	Followed by Problem 2
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1987 AIME Problems/Problem 1

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