2017 AMC 10B Problems/Problem 16
Problem
How many of the base-ten numerals for the positive integers less than or equal to 
contain the digit 
?
Solution 1
We can use complementary counting. There are positive integers in total to consider, and there are 
one-digit integers, 
two digit integers without a zero, 
three digit integers without a zero, and four-digit integers starting with a 1 without a zero. Therefore, the answer is 
.
Solution 2 (Casework)
We can use casework to solve this problem. First, we notice there are no one-digit numbers that contain a zero. There are two-digit integers and 
three-digit integers containing at least one zero. Next, we consider the four-digit integers beginning with one. There are 
of these four-digit integers with one zero, 
with two zeros, and 
with three zeros 
. Finally, we consider the numbers 
to which all contain at least one zero. Adding all of these together we get 
.
See Also
| 2017 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
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