Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2017 AMC 10B Problems/Problem 18"

m (→‎Solution)
m
Line 7: Line 7:
 
===Solution 1===
 
===Solution 1===
 
First we figure out the number of ways to put the <math>3</math> blue disks. Denote the spots to put the disks as <math>1-6</math> from left to right, top to bottom. The cases to put the blue disks are <math>(1,2,3),(1,2,4),(1,2,5),(1,2,6),(2,3,5),(1,4,6)</math>. For each of those cases we can easily figure out the number of ways for each case, so the total amount is <math>2+2+3+3+1+1 = \boxed{\textbf{(D) } 12}</math>.
 
First we figure out the number of ways to put the <math>3</math> blue disks. Denote the spots to put the disks as <math>1-6</math> from left to right, top to bottom. The cases to put the blue disks are <math>(1,2,3),(1,2,4),(1,2,5),(1,2,6),(2,3,5),(1,4,6)</math>. For each of those cases we can easily figure out the number of ways for each case, so the total amount is <math>2+2+3+3+1+1 = \boxed{\textbf{(D) } 12}</math>.
−
 
−
==Solution 2==
 
−
Denote the <math>6</math> discs as in the first solution. Ignoring reflections or rotations, there are <math>\binom{6}{3} * \binom{3}{2} = 60</math> colorings. Now we need to count the number of fixed points under possible transformations:
 
−
 
−
1. The identify transformation. Since this doesn't change anything, there are <math>60</math> fixed points
 
−
 
−
2. Reflect about a line of symmetry. There are <math>3</math> lines of reflections. Take the line of reflection going through the centers of circles <math>1</math> and <math>5</math>. Then, the colors of circles <math>2</math> and <math>3</math> must be the same, and the colors of circles <math>4</math> and <math>6</math> must be the same. This gives us <math>4</math> fixed points per line of reflection
 
−
 
−
3. Rotate by <math>120^\circ</math> counter clockwise or clockwise with respect to the center of the diagram. Take the clockwise case for example. There will be a fixed point in this case if the colors of circles <math>1</math>, <math>4</math>, and <math>6</math> will be the same. Similarly, the colors of circles <math>2</math>, <math>3</math>, and <math>5</math> will be the same. This is impossible, so this case gives us <math>0</math> fixed points per rotation.
 
−
 
−
By Burnside's lemma, the total number of colorings is <math>(1*60+3*4+2*0)/(1+3+2) = \boxed{\textbf{(D) } 12}</math>.
 
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2017|ab=B|num-b=17|num-a=19}}
 
{{AMC10 box|year=2017|ab=B|num-b=17|num-a=19}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 19:02, 16 February 2017

Problem

In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ 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?

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 15$

Solution

Solution 1

First we figure out the number of ways to put the $3$ blue disks. Denote the spots to put the disks as $1-6$ from left to right, top to bottom. The cases to put the blue disks are $(1,2,3),(1,2,4),(1,2,5),(1,2,6),(2,3,5),(1,4,6)$. For each of those cases we can easily figure out the number of ways for each case, so the total amount is $2+2+3+3+1+1 = \boxed{\textbf{(D) } 12}$.

See Also

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_10B_Problems/Problem_18&oldid=83879"