Difference between revisions of "2006 AMC 10A Problems/Problem 21"
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== Problem == | == Problem == | ||
| − | How many four-digit positive | + | How many four-[[digit]] [[positive integer]]s have at least one digit that is a 2 or a 3? |
<math>\mathrm{(A) \ } 2439\qquad\mathrm{(B) \ } 4096\qquad\mathrm{(C) \ } 4903\qquad\mathrm{(D) \ } 4904\qquad\mathrm{(E) \ } 5416\qquad</math> | <math>\mathrm{(A) \ } 2439\qquad\mathrm{(B) \ } 4096\qquad\mathrm{(C) \ } 4903\qquad\mathrm{(D) \ } 4904\qquad\mathrm{(E) \ } 5416\qquad</math> | ||
| − | |||
| − | + | == Video Solution == | |
| + | https://youtu.be/0W3VmFp55cM?t=3291 | ||
| − | + | ~ pi_is_3.14 | |
| − | + | == Solution (Complementary Counting) == | |
| + | Since we are asked for the number of positive 4-digit [[integer]]s with at least 2 or 3 in it, we can find this by finding the total number of 4-digit integers and subtracting off those which do not have any 2s or 3s as digits. | ||
| − | + | The total number of 4-digit integers is <math>9 \cdot 10 \cdot 10 \cdot 10 = 9000</math>, since we have 10 choices for each digit except the first (which can't be 0). | |
| + | Similarly, the total number of 4-digit integers without any 2 or 3 is <math>7 \cdot 8 \cdot 8 \cdot 8 ={3584}</math>. | ||
| − | == See | + | Therefore, the total number of positive 4-digit integers that have at least one 2 or 3 in their decimal representation is <math>9000-3584=\boxed{5416} \Longrightarrow \mathrm{(E)} </math> |
| − | + | ||
| − | + | == See also == | |
| + | {{AMC10 box|year=2006|ab=A|num-b=20|num-a=22}} | ||
| + | |||
| + | [[Category:Introductory Combinatorics Problems]] | ||
| + | {{MAA Notice}} | ||
Revision as of 23:26, 16 January 2021
Problem
How many four-digit positive integers have at least one digit that is a 2 or a 3?
Video Solution
https://youtu.be/0W3VmFp55cM?t=3291
~ pi_is_3.14
Solution (Complementary Counting)
Since we are asked for the number of positive 4-digit integers with at least 2 or 3 in it, we can find this by finding the total number of 4-digit integers and subtracting off those which do not have any 2s or 3s as digits.
The total number of 4-digit integers is 
, since we have 10 choices for each digit except the first (which can't be 0).
Similarly, the total number of 4-digit integers without any 2 or 3 is 
.
Therefore, the total number of positive 4-digit integers that have at least one 2 or 3 in their decimal representation is 
See also
| 2006 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 20 |
Followed by Problem 22 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
