Difference between revisions of "2005 AMC 12A Problems/Problem 23"
(→Problem) |
Armalite46 (talk | contribs) m (→Solution) |
||
(6 intermediate revisions by 4 users not shown) | |||
Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
− | Two distinct [[ | + | Two distinct [[number]]s a and b are chosen randomly from the [[set]] <math>\{2, 2^2, 2^3, ..., 2^{25}\}</math>. What is the [[probability]] that <math>\mathrm{log}_a b</math> is an [[integer]]? |
+ | |||
+ | <math>\mathrm{(A)}\ \frac{2}{25}\qquad \mathrm{(B)}\ \frac{31}{300}\qquad \mathrm{(C)}\ \frac{13}{100}\qquad \mathrm{(D)}\ \frac{7}{50}\qquad \mathrm{(E)}\ \frac 12</math> | ||
== Solution == | == Solution == | ||
− | {{ | + | Let <math>\log_a b = z</math>, so <math>a^z = b</math>. Define <math>a = 2^x</math>, <math>b = 2^y</math>; then <math>\left(2^x\right)^z = 2^{xz}= 2^y</math>, so <math>x|y</math>. Here we can just make a table and count the number of values of <math>y</math> per value of <math>x</math>. The largest possible value of <math>x</math> is 12, and we get <math>\sum_{x=1}^{12} \lfloor\frac {25}x-1\rfloor = 24 + 11 + 7 + 5 + 4 + 3 + 2 + 2 + 1 + 1 + 1 + 1 = 62</math>. |
+ | |||
+ | The total number of ways to pick two distinct numbers is <math>\frac{25!}{(25-2)!}= 25 \cdot 24 = 600</math>, so we get a probability of <math>\frac{62}{600} = \boxed{\textbf{(B) }\frac{31}{300}}</math>. | ||
+ | |||
== See also == | == See also == | ||
− | + | {{AMC12 box|year=2005|ab=A|num-b=22|num-a=24}} | |
− | + | ||
− | + | [[Category:Introductory Combinatorics Problems]] | |
+ | {{MAA Notice}} |
Latest revision as of 19:35, 7 November 2013
Problem
Two distinct numbers a and b are chosen randomly from the set . What is the probability that is an integer?
Solution
Let , so . Define , ; then , so . Here we can just make a table and count the number of values of per value of . The largest possible value of is 12, and we get .
The total number of ways to pick two distinct numbers is , so we get a probability of .
See also
2005 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.