Question

How many of the base-ten numerals for the positive integers less than or equal to $2017$ contain the digit $0?$

$469$

$471$

$475$

$478$

$481$

Solution:

For numbers less than $100,$ we only have a $0$ if it is a multiple of $10,$ of which there are $9.$

For numbers between $100$ and $999$ inclusive, we will use complementary counting. There are $900$ total numbers in this range. Also, there are $9\cdot 9\cdot 9=729$ numbers in this range with no $0$ since there are $9$ ways to choose each digit to not be $0.$ Thus, the total in this range is $171.$

For numbers between $1000$ and $1999$ inclusive, we will use complementary counting again. There are $1000$ total numbers in this range. Also, there are $1\cdot 9\cdot 9\cdot 9=729$ numbers in this range with no $0$ since there are $9$ ways to choose each of the last $3$ digits to not be $0$ and the first digit must be $1.$ Thus, the total in this range is $271.$

There are $18$ numbers between $2000$ and $2017$ inclusive, each with a $0$ in the second digit from the left.

This makes the total $9+171+271+18=469.$

Thus, the correct answer is A .

2017 AMC 10B Problem 16

Solution:

Problem 16 in Other Years