Difference between revisions of "2010 AMC 12B Problems/Problem 16"
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{{duplicate|[[2010 AMC 12B Problems|2010 AMC 12B #16]] and [[2010 AMC 10B Problems|2010 AMC 10B #18]]}} | {{duplicate|[[2010 AMC 12B Problems|2010 AMC 12B #16]] and [[2010 AMC 10B Problems|2010 AMC 10B #18]]}} | ||
− | == Problem | + | == Problem == |
Positive integers <math>a</math>, <math>b</math>, and <math>c</math> are randomly and independently selected with replacement from the set <math>\{1, 2, 3,\dots, 2010\}</math>. What is the probability that <math>abc + ab + a</math> is divisible by <math>3</math>? | Positive integers <math>a</math>, <math>b</math>, and <math>c</math> are randomly and independently selected with replacement from the set <math>\{1, 2, 3,\dots, 2010\}</math>. What is the probability that <math>abc + ab + a</math> is divisible by <math>3</math>? | ||
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If <math>3|a</math>, we are done. There is a probability of <math>\frac{1}{3}</math> that that happens. | If <math>3|a</math>, we are done. There is a probability of <math>\frac{1}{3}</math> that that happens. | ||
− | Otherwise, we have <math>3|bc+b+1</math>, which means that <math>b(c+1) \equiv 2\pmod{3}</math>. So either <cmath>b \equiv 1 \pmod{3}, c \equiv 1 \pmod{3}</cmath> or <cmath>b \equiv 2 \pmod {3}, c \equiv 0 \pmod 3</cmath> which will lead to the property being true. There | + | Otherwise, we have <math>3|bc+b+1</math>, which means that <math>b(c+1) \equiv 2\pmod{3}</math>. So either <cmath>b \equiv 1 \pmod{3}, c \equiv 1 \pmod{3}</cmath> or <cmath>b \equiv 2 \pmod {3}, c \equiv 0 \pmod 3</cmath> which will lead to the property being true. There is a <math>\frac{1}{3}\cdot\frac{1}{3}=\frac{1}{9}</math> chance for each bundle of cases to be true. Thus, the total for the cases is <math>\frac{2}{9}</math>. But we have to multiply by <math>\frac{2}{3}</math> because this only happens with a <math>\frac{2}{3}</math> chance. So the total is actually <math>\frac{4}{27}</math>. |
The grand total is <cmath>\frac{1}{3} + \frac{4}{27} = \boxed{\text{(E) }\frac{13}{27}.}</cmath> | The grand total is <cmath>\frac{1}{3} + \frac{4}{27} = \boxed{\text{(E) }\frac{13}{27}.}</cmath> | ||
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Just like solution 1, we see that there is a <math>\frac{1}{3}</math> chance of <math>3|a</math> and <math>\frac{2}{9}</math> chance of <math>3|1+b+bc</math> | Just like solution 1, we see that there is a <math>\frac{1}{3}</math> chance of <math>3|a</math> and <math>\frac{2}{9}</math> chance of <math>3|1+b+bc</math> | ||
− | Now, we can just use PIE ( | + | Now, we can just use PIE (Principles of Inclusion and Exclusion) to get our answer to be <math>\frac{1}{3}+\frac{2}{9}-\frac{1}{3}\cdot\frac{2}{9} = \boxed{\text{(E) } \frac{13}{27}}</math> |
-Conantwiz2023 | -Conantwiz2023 | ||
+ | |||
+ | ==Solution 3 (Fancier version of Solution 1)== | ||
+ | |||
+ | As with solution one, we conclude that if <math>a\equiv0\mod 3</math> then the requirements are satisfied. We then have: | ||
+ | <cmath>a(bc+c)+a\equiv0 \mod 3</cmath> | ||
+ | <cmath>a(bc+c)\equiv-a \mod 3</cmath> | ||
+ | <cmath>c(b+1)\equiv-1 \mod 3</cmath> | ||
+ | <cmath>b+1\equiv \frac{-1}{c} \mod 3</cmath> | ||
+ | Which is true for one <math>b</math> when <math>c\not\equiv 0 \mod 3</math> because the integers<math>\mod 3</math> form a field under multiplication and addition with absorbing element <math>0</math>. | ||
+ | |||
+ | This gives us <math>P=\frac{1}{3}+\frac{4}{27}=\boxed{\text{(E) }\frac{13}{27}}</math>. | ||
+ | |||
+ | ~Snacc | ||
==Video Solution== | ==Video Solution== | ||
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== See also == | == See also == | ||
{{AMC12 box|year=2010|num-b=15|num-a=17|ab=B}} | {{AMC12 box|year=2010|num-b=15|num-a=17|ab=B}} | ||
+ | {{AMC10 box|year=2010|num-b=17|num-a=19|ab=B}} | ||
[[Category:Introductory Combinatorics Problems]] | [[Category:Introductory Combinatorics Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 18:56, 22 June 2024
- The following problem is from both the 2010 AMC 12B #16 and 2010 AMC 10B #18, so both problems redirect to this page.
Contents
Problem
Positive integers ,
, and
are randomly and independently selected with replacement from the set
. What is the probability that
is divisible by
?
Solution 1
We group this into groups of , because
. This means that every residue class mod 3 has an equal probability.
If , we are done. There is a probability of
that that happens.
Otherwise, we have , which means that
. So either
or
which will lead to the property being true. There is a
chance for each bundle of cases to be true. Thus, the total for the cases is
. But we have to multiply by
because this only happens with a
chance. So the total is actually
.
The grand total is
Solution 2 (Minor change from Solution 1)
Just like solution 1, we see that there is a chance of
and
chance of
Now, we can just use PIE (Principles of Inclusion and Exclusion) to get our answer to be
-Conantwiz2023
Solution 3 (Fancier version of Solution 1)
As with solution one, we conclude that if then the requirements are satisfied. We then have:
Which is true for one
when
because the integers
form a field under multiplication and addition with absorbing element
.
This gives us .
~Snacc
Video Solution
https://youtu.be/FQO-0E2zUVI?t=437
~IceMatrix
See also
2010 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 15 |
Followed by Problem 17 |
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 |
2010 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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.