Difference between revisions of "2002 AMC 8 Problems/Problem 19"
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<math> \text{(A)}\ 72\qquad\text{(B)}\ 90\qquad\text{(C)}\ 144\qquad\text{(D)}\ 162\qquad\text{(E)}\ 180 </math> | <math> \text{(A)}\ 72\qquad\text{(B)}\ 90\qquad\text{(C)}\ 144\qquad\text{(D)}\ 162\qquad\text{(E)}\ 180 </math> | ||
| − | ==Solution== | + | ==Solution 1== |
Numbers with exactly one zero have the form <math>\overline{a0b}</math> or <math>\overline{ab0}</math>, where the <math>a,b \neq 0</math>. There are <math>(9\cdot1\cdot9)+(9\cdot9\cdot1) = 81+81 = \boxed{162}</math> such numbers, hence our answer is <math>\fbox{D}</math>. | Numbers with exactly one zero have the form <math>\overline{a0b}</math> or <math>\overline{ab0}</math>, where the <math>a,b \neq 0</math>. There are <math>(9\cdot1\cdot9)+(9\cdot9\cdot1) = 81+81 = \boxed{162}</math> such numbers, hence our answer is <math>\fbox{D}</math>. | ||
Latest revision as of 23:54, 7 December 2024
Contents
Problem
How many whole numbers between 99 and 999 contain exactly one 0?
Solution 1
Numbers with exactly one zero have the form 
or 
, where the 
. There are 
such numbers, hence our answer is 
.
Solution 2 (Complementary Counting)
(Whole numbers between 99 and 999)-(Whole numbers which do not contain exactly one 0)= (How many whole numbers between 99 and 999 contain exactly one 0).
How many whole numbers are between 99 and 999, 
, so there are 900 numbers between 99 and 999.
How many whole numbers in this range do not contain the digit 
, there are 
possible digits for each digit in this three digit number, 
. So there are 
numbers in this range which do not contain the digit .
How many whole numbers in this range contain the digit 2 times, there are possible digits for the first digit and the other two digits have to be the digit . So there are 
of these numbers.
So there are 
whole numbers between 99 and 999 that contain exactly one 0.
Simplifying the expression we get that there are 
whole numbers between 99 and 999 that contain exactly one 0. So the answer is .
~blankbox
Video Solution
https://youtu.be/nctNL-xLImI Soo, DRMS, NM
https://www.youtube.com/watch?v=eAeVBrQ1PQI ~David
Video Solution by WhyMath
See Also
| 2002 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 18 |
Followed by Problem 20 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
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