Difference between revisions of "2017 AMC 12B Problems/Problem 13"
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− | ==Problem | + | ==Problem== |
In the figure below, <math>3</math> of the <math>6</math> disks are to be painted blue, <math>2</math> are to be painted red, and <math>1</math> is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible? | In the figure below, <math>3</math> of the <math>6</math> disks are to be painted blue, <math>2</math> are to be painted red, and <math>1</math> is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible? | ||
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<math>\textbf{(A) } 6 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 15</math> | <math>\textbf{(A) } 6 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 15</math> | ||
− | ==Solution== | + | ==Solution 1== |
− | Looking at the answer choices, we see that the possibilities are indeed countable. Thus, we will utilize that approach in the form of two separate cases, as rotation and reflection take care of numerous possibilities. First, consider the case that the green | + | Looking at the answer choices, we see that the possibilities are indeed countable. Thus, we will utilize that approach in the form of two separate cases, as rotation and reflection take care of numerous possibilities. First, consider the case that the green disk is in a corner. This yields <math>6</math> possible arrangements for the <math>3</math> blue disks and <math>2</math> red disks in the remaining available slots. Now, consider the case that the green disk is on an edge. This yields <math>6</math> more possible arrangements for the <math>3</math> blue disks and <math>2</math> red disks in the remaining available slots. Thus, our answer is <math>6 + 6 = \boxed{\bold{(D)}\, 12}</math> |
Solution by akaashp11 | Solution by akaashp11 | ||
+ | |||
+ | ==Solution 2 (Burnside's Lemma)== | ||
+ | Burnside's Lemma can be applied to the problem. Number the disks 1-6 from top to bottom and left to right. First, we count the number of group actions. There are <math>6</math> in total: <math>3</math> rotations, <math>0</math>°, <math>120</math>°, and <math>240</math>°. Additionally, there are <math>3</math> different reflections, which are over lines passing through the middle of disks <math>1</math>, <math>4</math>, and <math>6</math>. The <math>0</math>° rotation has the total number of diagrams fixed, which is <math>6!/3!\cdot2!=60</math>. The <math>120</math>° and <math>240</math>° rotations each have <math>0</math> fixed diagrams, as one would need the same color for disks | ||
+ | <math>1, 4, 6</math> and the same color for disks <math>2, 3, 5,</math> which isn't possible. Consider the reflection over the line passing through disk <math>1</math>. | ||
+ | The green can be placed in <math>2</math> places, disk <math>1</math> or <math>5</math>, and then, the blue must go in either disks <math>2</math> and <math>3</math> or disks <math>4</math> and <math>6</math>, so the number of fixed diagrams is <math>2\cdot2=4</math>. The other two reflections also have <math>4</math> unique fixed diagrams. Therefore, applying Burnside's Lemma, we get <math>(60+4+4+4)/6 = \boxed{\bold{(D)}\, 12}</math> | ||
+ | |||
+ | Solution by Aadileo | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/T7eNQISBzsc?si=eOKSzNorVrHaV1px ~TheBeautyOfMath | ||
+ | |||
+ | -aashrithm | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2017|ab=B|num-b=12|num-a=14}} | {{AMC12 box|year=2017|ab=B|num-b=12|num-a=14}} | ||
+ | {{AMC10 box|year=2017|ab=B|num-b=17|num-a=19}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
+ | |||
+ | [[Category:Introductory Combinatorics Problems]] |
Latest revision as of 10:00, 28 July 2024
Problem
In the figure below, of the
disks are to be painted blue,
are to be painted red, and
is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
Solution 1
Looking at the answer choices, we see that the possibilities are indeed countable. Thus, we will utilize that approach in the form of two separate cases, as rotation and reflection take care of numerous possibilities. First, consider the case that the green disk is in a corner. This yields possible arrangements for the
blue disks and
red disks in the remaining available slots. Now, consider the case that the green disk is on an edge. This yields
more possible arrangements for the
blue disks and
red disks in the remaining available slots. Thus, our answer is
Solution by akaashp11
Solution 2 (Burnside's Lemma)
Burnside's Lemma can be applied to the problem. Number the disks 1-6 from top to bottom and left to right. First, we count the number of group actions. There are in total:
rotations,
°,
°, and
°. Additionally, there are
different reflections, which are over lines passing through the middle of disks
,
, and
. The
° rotation has the total number of diagrams fixed, which is
. The
° and
° rotations each have
fixed diagrams, as one would need the same color for disks
and the same color for disks
which isn't possible. Consider the reflection over the line passing through disk
.
The green can be placed in
places, disk
or
, and then, the blue must go in either disks
and
or disks
and
, so the number of fixed diagrams is
. The other two reflections also have
unique fixed diagrams. Therefore, applying Burnside's Lemma, we get
Solution by Aadileo
Video Solution
https://youtu.be/T7eNQISBzsc?si=eOKSzNorVrHaV1px ~TheBeautyOfMath
-aashrithm
See Also
2017 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
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 |
2017 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 |
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