Difference between revisions of "2015 AMC 10B Problems/Problem 11"
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==Solution 2 (Listing)== | ==Solution 2 (Listing)== | ||
Since the only primes digits are <math>2</math>, <math>3</math>, <math>5</math>, and <math>7</math>, it doesn't seem too hard to list all of the numbers out. | Since the only primes digits are <math>2</math>, <math>3</math>, <math>5</math>, and <math>7</math>, it doesn't seem too hard to list all of the numbers out. | ||
− | + | ||
− | + | *2- Prime; | |
− | + | *3- Prime; | |
− | + | *5- Prime; | |
− | + | *7- Prime; | |
− | + | *22- Composite; | |
− | + | *23- Prime; | |
− | + | *25- Composite; | |
− | + | *27- Composite; | |
− | + | *32- Composite; | |
− | + | *33- Composite; | |
− | + | *35- Composite; | |
− | + | *37- Prime; | |
− | + | *52- Composite; | |
− | + | *53- Prime; | |
− | + | *55- Composite; | |
− | + | *57- Composite; | |
− | + | *72- Composite; | |
− | + | *73- Prime; | |
− | + | *75- Composite; | |
+ | *77- Composite. | ||
+ | |||
Counting it out, there are <math>20</math> cases and <math>8</math> of these are prime. So the answer is <math>\dfrac{8}{20}=\boxed{\textbf{(B)} \dfrac{2}{5}}</math>. | Counting it out, there are <math>20</math> cases and <math>8</math> of these are prime. So the answer is <math>\dfrac{8}{20}=\boxed{\textbf{(B)} \dfrac{2}{5}}</math>. | ||
~JH. L | ~JH. L |
Revision as of 22:30, 16 June 2022
Problem
Among the positive integers less than , each of whose digits is a prime number, one is selected at random. What is the probability that the selected number is prime?
Solution 1
The one digit prime numbers are , , , and . So there are a total of ways to choose a two digit number with both digits as primes and ways to choose a one digit prime, for a total of ways. Out of these , , , , , , , and are prime. Thus the probability is .
Solution 2 (Listing)
Since the only primes digits are , , , and , it doesn't seem too hard to list all of the numbers out.
- 2- Prime;
- 3- Prime;
- 5- Prime;
- 7- Prime;
- 22- Composite;
- 23- Prime;
- 25- Composite;
- 27- Composite;
- 32- Composite;
- 33- Composite;
- 35- Composite;
- 37- Prime;
- 52- Composite;
- 53- Prime;
- 55- Composite;
- 57- Composite;
- 72- Composite;
- 73- Prime;
- 75- Composite;
- 77- Composite.
Counting it out, there are cases and of these are prime. So the answer is . ~JH. L
Video Solution
~savannahsolver
See Also
2015 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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.