Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 19"
(→Solution) |
(→Solution) |
||
(2 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? | 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>[[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> | <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]] | ||
+ | |||
− | + | [[Category:Introductory Combinatorics Problems]] | |
− |
Latest revision as of 16:37, 17 August 2006
Problem
In the figure below, there are 4 distinct dots and , 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?
Solution
There are 4 color choices for dot . After coloring dot , there are 3 color choices for dot . If dot is the same color as dot (1 way), there are 3 choices for dot . If dot is a different color from dot (2 ways), there are only 2 choices for dot . Thus we have in total possible colorings, so choice is the answer.