Difference between revisions of "2015 AMC 8 Problems/Problem 10"
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| + | ==Problem== | ||
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How many integers between <math>1000</math> and <math>9999</math> have four distinct digits? | How many integers between <math>1000</math> and <math>9999</math> have four distinct digits? | ||
<math>\textbf{(A) }3024\qquad\textbf{(B) }4536\qquad\textbf{(C) }5040\qquad\textbf{(D) }6480\qquad \textbf{(E) }6561</math> | <math>\textbf{(A) }3024\qquad\textbf{(B) }4536\qquad\textbf{(C) }5040\qquad\textbf{(D) }6480\qquad \textbf{(E) }6561</math> | ||
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==Solution 1== | ==Solution 1== | ||
| − | + | There are <math>9</math> choices for the first number, since it cannot be <math>0</math>, there are only <math>9</math> choices left for the second number since it must differ from the first, <math>8</math> choices for the third number, since it must differ from the first two, and <math>7</math> choices for the fourth number, since it must differ from all three. This means there are <math>9 \times 9 \times 8 \times 7=\boxed{\textbf{(B) }4536}</math> integers between <math>1000</math> and <math>9999</math> with four distinct digits. | |
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| + | If you're too lazy to do <math>9 \times 9 \times 8 \times 7= 4536</math>, we can notice <math> 9 \times 9 = 81</math> and <math> 8 \times 7=56</math> The unit digit must be 6. Therefore the answer is the only answer with a a unit digit of 6. <math>\boxed{B}</math>) | ||
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| + | ==Video Solution (HOW TO THINK CRITICALLY!!!)== | ||
| + | https://youtu.be/-VcuRyDB_wI | ||
| + | |||
| + | ~Education, the Study of Everything | ||
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| + | ==Video solution== | ||
| + | https://youtu.be/Zhsb5lv6jCI?t=272 | ||
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| + | https://www.youtube.com/watch?v=OESYIYjZFdk ~David | ||
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| + | https://youtu.be/2nfFg8JXKFE | ||
| + | |||
| + | ~savannahsolver | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2015|num-b=9|num-a=11}} | {{AMC8 box|year=2015|num-b=9|num-a=11}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 09:10, 18 August 2024
Contents
Problem
How many integers between 
and 
have four distinct digits?
Solution 1
There are 
choices for the first number, since it cannot be 
, there are only choices left for the second number since it must differ from the first, 
choices for the third number, since it must differ from the first two, and 
choices for the fourth number, since it must differ from all three. This means there are 
integers between and with four distinct digits.
If you're too lazy to do 
, we can notice 
and 
The unit digit must be 6. Therefore the answer is the only answer with a a unit digit of 6. 
)
Video Solution (HOW TO THINK CRITICALLY!!!)
~Education, the Study of Everything
Video solution
https://youtu.be/Zhsb5lv6jCI?t=272
https://www.youtube.com/watch?v=OESYIYjZFdk ~David
~savannahsolver
See Also
| 2015 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 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. 
