Difference between revisions of "2004 AMC 8 Problems/Problem 2"
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<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 81 </math> | <math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 81 </math> | ||
− | == Solution == | + | |
+ | == Solution 1 == | ||
+ | We can solve this problem easily, just by calculating how many choices there are for each of the four digits. | ||
+ | First off, we know there are only <math>2</math> choices for the first digit, because <math>0</math> isn't a valid choice, or the number would a 3-digit number, which is not what we want. | ||
+ | We have <math>3</math> choices for the second digit, since we already used up one of the digits, and <math>2</math> choices for the third, and finally just <math>1</math> choices for the fourth and final one. | ||
+ | <math>2*3*2*1</math> is 12, but there are 2 zeros that have been counted as different numbers, so divide by 2 to get <math>\boxed{\textbf{(B)}\ 6}</math>. | ||
+ | |||
+ | == Solution 2== | ||
Note that the four-digit number must start with either a <math>2</math> or a <math>4</math>. The four-digit numbers that start with <math>2</math> are <math>2400, 2040</math>, and <math>2004</math>. The four-digit numbers that start with <math>4</math> are <math>4200, 4020</math>, and <math>4002</math> which gives us a total of <math>\boxed{\textbf{(B)}\ 6}</math>. | Note that the four-digit number must start with either a <math>2</math> or a <math>4</math>. The four-digit numbers that start with <math>2</math> are <math>2400, 2040</math>, and <math>2004</math>. The four-digit numbers that start with <math>4</math> are <math>4200, 4020</math>, and <math>4002</math> which gives us a total of <math>\boxed{\textbf{(B)}\ 6}</math>. | ||
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==See Also== | ==See Also== | ||
{{AMC8 box|year=2004|num-b=1|num-a=3}} | {{AMC8 box|year=2004|num-b=1|num-a=3}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 23:00, 13 July 2019
Contents
Problem
How many different four-digit numbers can be formed be rearranging the four digits in ?
Solution 1
We can solve this problem easily, just by calculating how many choices there are for each of the four digits. First off, we know there are only choices for the first digit, because isn't a valid choice, or the number would a 3-digit number, which is not what we want. We have choices for the second digit, since we already used up one of the digits, and choices for the third, and finally just choices for the fourth and final one. is 12, but there are 2 zeros that have been counted as different numbers, so divide by 2 to get .
Solution 2
Note that the four-digit number must start with either a or a . The four-digit numbers that start with are , and . The four-digit numbers that start with are , and which gives us a total of .
See Also
2004 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.