Difference between revisions of "2006 AIME A Problems/Problem 7"
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== Problem == | == Problem == | ||
| − | Find the number of ordered | + | Find the number of [[ordered pair]]s of [[positive]] [[integer]]s <math> (a,b) </math> such that <math> a+b=1000 </math> and neither <math> a </math> nor <math> b </math> has a [[zero]] [[digit]]. |
== Solution == | == Solution == | ||
| + | There are <math>\frac{1000}{10} = 100</math> numbers up to 1000 that have 0 as their units digit. All of the other excluded possibilities for <math>a</math> are when <math>a</math> or <math>b</math> have a 0 in the tens digit, or if the tens digit of <math>a</math> is 0 or 9. Excluding the hundreds, which were counted above already, there are <math>9 \cdot 2 = 18</math> numbers in every hundred numbers that have a tens digit of 0 or 9, totaling <math>10 \cdot 18 = 180</math> such numbers. However, the numbers from 1 to 9 and 991 to 999 do not have 0s, so we must subtract <math>18</math> from that to get <math>162</math>. Therefore, there are <math>1000 - (100 + 162) = 738</math> such ordered pairs. | ||
== See also == | == See also == | ||
| − | + | {{AIME box|year=2006|n=II|num-b=6|num-a=8}} | |
[[Category:Intermediate Combinatorics Problems]] | [[Category:Intermediate Combinatorics Problems]] | ||
Revision as of 20:27, 21 February 2007
Problem
Find the number of ordered pairs of positive integers 
such that 
and neither 
nor 
has a zero digit.
Solution
There are 
numbers up to 1000 that have 0 as their units digit. All of the other excluded possibilities for are when or have a 0 in the tens digit, or if the tens digit of is 0 or 9. Excluding the hundreds, which were counted above already, there are 
numbers in every hundred numbers that have a tens digit of 0 or 9, totaling 
such numbers. However, the numbers from 1 to 9 and 991 to 999 do not have 0s, so we must subtract 
from that to get 
. Therefore, there are 
such ordered pairs.
See also
| 2006 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
