2005 AMC 12A Problems/Problem 19
Problem
A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled?
Solutions
Solution 1
We find the number of numbers with a and subtract from . Quick counting tells us that there are numbers with a 4 in the hundreds place, numbers with a 4 in the tens place, and numbers with a 4 in the units place (counting ). Now we apply the Principle of Inclusion-Exclusion. There are numbers with a 4 in the hundreds and in the tens, and for both the other two intersections. The intersection of all three sets is just . So we get:
Solution 2
Alternatively, consider that counting without the number is almost equivalent to counting in base ; only, in base , the number is not counted. Since is skipped, the symbol represents miles of travel, and we have traveled miles. By basic conversion,
Solution 3
Since any numbers containing one or more s were skipped, we need only to find the numbers that don't contain a at all. First we consider - . Single digits are simply four digit numbers with a zero in all but the ones place (this concept applies to double and triple digits numbers as well). From - , we have possibilities for the thousands place, and possibilities for the hundreds, tens, and ones places. This is possibilities (because doesn't count) or numbers. From - there are numbers, of which don't contain a . Therefore the total is , or .
See also
2005 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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