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 "University of South Carolina High School Math Contest/1993 Exam/Problem 19"

 
(→‎Solution)
 
(4 intermediate revisions by one other user not shown)
Line 1: Line 1:
 
== Problem ==
 
== Problem ==
 +
In the figure below, there are 4 distinct dots <math>A, B, C,</math> and <math>D</math>, joined by edges.  Each dot is to be colored either red, blue, green, or yellow.  No two dots joined by an edge are to be colored with the same color.  How many completed colorings are possible?
  
−
<center><math> \mathrm{(A) \ } \qquad \mathrm{(B) \ } \qquad \mathrm{(C) \ } \qquad \mathrm{(D) \ } \qquad \mathrm{(E) \ }  </math></center>
+
<center>[[Image:Usc93.19.PNG]]</center>
 +
 
 +
<center><math> \mathrm{(A) \ }24 \qquad \mathrm{(B) \ }72 \qquad \mathrm{(C) \ }84 \qquad \mathrm{(D) \ }96 \qquad \mathrm{(E) \ }108 </math></center>
  
 
== Solution ==
 
== Solution ==
 +
There are 4 color choices for dot <math>A</math>.  After coloring dot <math>A</math>, there are 3 color choices for dot <math>B</math>.    If dot <math>D</math> is the same color as dot <math>B</math> (1 way), there are 3 choices for dot <math>C</math>.  If dot <math>D</math> is a different color from dot <math>B</math> (2 ways), there are only 2 choices for dot <math>C</math>.  Thus we have in total <math>4\cdot3\cdot(1\cdot3 + 2\cdot2) = 84</math> possible colorings, so choice <math>\mathrm{(C)}</math> is the answer.
 +
 +
----
 +
 +
* [[University of South Carolina High School Math Contest/1993 Exam/Problem 18|Previous Problem]]
 +
* [[University of South Carolina High School Math Contest/1993 Exam/Problem 20|Next Problem]]
 +
* [[University of South Carolina High School Math Contest/1993 Exam|Back to Exam]]
 +
  
−
== See also ==
+
[[Category:Introductory Combinatorics Problems]]
−
* [[University of South Carolina High School Math Contest/1993 Exam]]
 

Latest revision as of 16:37, 17 August 2006

Problem

In the figure below, there are 4 distinct dots $A, B, C,$ and $D$, joined by edges. Each dot is to be colored either red, blue, green, or yellow. No two dots joined by an edge are to be colored with the same color. How many completed colorings are possible?

Usc93.19.PNG
$\mathrm{(A) \ }24 \qquad \mathrm{(B) \ }72 \qquad \mathrm{(C) \ }84 \qquad \mathrm{(D) \ }96 \qquad \mathrm{(E) \ }108$

Solution

There are 4 color choices for dot $A$. After coloring dot $A$, there are 3 color choices for dot $B$. If dot $D$ is the same color as dot $B$ (1 way), there are 3 choices for dot $C$. If dot $D$ is a different color from dot $B$ (2 ways), there are only 2 choices for dot $C$. Thus we have in total $4\cdot3\cdot(1\cdot3 + 2\cdot2) = 84$ possible colorings, so choice $\mathrm{(C)}$ is the answer.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=University_of_South_Carolina_High_School_Math_Contest/1993_Exam/Problem_19&oldid=9629"