2017 AMC 8 Problems/Problem 20: Difference between revisions
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==Problem== | ==Problem== | ||
An integer between 1000 and 9999, inclusive, is chosen at random. What is the probability that it | An integer between 1000 and 9999, inclusive, is chosen at random. What is the probability that it | ||
is an odd integer whose digits are all distinct? <math>(A) 14 | is an odd integer whose digits are all distinct? | ||
<math>\textbf{(A) }\frac{14}{75} \qquad \textbf{(B) }\frac{56}{225} \qquad \textbf{(C) }\frac{107}{400} \qquad \textbf{(D) }\frac{7}{25} \qquad \textbf{(E) }\frac{9}{25}</math> | |||
==Solution== | ==Solution== | ||
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{{MAA Notice}} | {{MAA Notice}} | ||
[[Category:Introductory Combinatorics Problems]] | |||
[[Category:Introductory Probability Problems]] | |||
Latest revision as of 23:49, 12 September 2025
Problem
An integer between 1000 and 9999, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct?
Solution
There are 
options for the last digit as the integer must be odd. The first digit now has 
options left (it can't be 
or the same as the last digit). The second digit also has options left (it can't be the same as the first or last digit). Finally, the third digit has 
options (it can't be the same as the three digits that are already chosen).
Since there are 
total integers, our answer is ![\[\frac{8 \cdot 8 \cdot 7 \cdot 5}{9000} = \boxed{\textbf{(B)}\ \frac{56}{225}}.\]](http://latex.artofproblemsolving.com/5/d/8/5d8a4f26914ee2febf86c13a27c746c91b00f952.png)
Video Solution (CREATIVE THINKING + ANALYSIS!!!)
~Education, the Study of Everything
Video Solution
https://youtu.be/tJm9KqYG4fU?t=3114
~savannahsolver
https://www.youtube.com/watch?v=2G9jiu5y5PM ~David
See Also
| 2017 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 19 |
Followed by Problem 21 | |
| 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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination
