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Difference between revisions of "2011 AMC 10A Problems/Problem 22"

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Case <math>1</math>: There are <math>6!</math> ways of assigning each vertex a different color. <math>6! = 720</math>
 
Case <math>1</math>: There are <math>6!</math> ways of assigning each vertex a different color. <math>6! = 720</math>
  
−
Case <math>2</math>: There are <math>\frac {6!}{2!} * 5</math> ways. After picking four colors, we can rotate our hexagon in <math>5</math> ways to get different outcomes. <math>\frac {6!}{2} * 5 = 1800</math>
+
Case <math>2</math>: There are <math>\frac {6!}{2!} * 5</math> ways. After picking four colors, we can rotate our pentagon in <math>5</math> ways to get different outcomes. <math>\frac {6!}{2} * 5 = 1800</math>
  
−
Case <math>3</math>: There are <math>\frac {\frac {6!}{3!} * 10}{2!}</math> ways of arranging the final case. We can pick <math>3</math> colors for our hexagon. There are <math>5</math> spots for the first pair of colors. Then, there are <math>2</math> possible ways we can put the final pair in the last <math>3</math> spaces. But because the two pairs are indistinguishable, we divide by <math>2!</math>. <math>\frac {\frac {6!}{3!} * 10}{2!} = 600</math>
+
Case <math>3</math>: There are <math>\frac {\frac {6!}{3!} * 10}{2!}</math> ways of arranging the final case. We can pick <math>3</math> colors for our pentagon. There are <math>5</math> spots for the first pair of colors. Then, there are <math>2</math> possible ways we can put the final pair in the last <math>3</math> spaces. But because the two pairs are indistinguishable, we divide by <math>2!</math>. <math>\frac {\frac {6!}{3!} * 10}{2!} = 600</math>
  
 
Adding all the possibilities up, we get <math>720+1800+600=\boxed{3120 \ \mathbf{(C)}}</math>
 
Adding all the possibilities up, we get <math>720+1800+600=\boxed{3120 \ \mathbf{(C)}}</math>

Revision as of 19:23, 3 August 2018

Problem 22

Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?

$\textbf{(A)}\ 2520\qquad\textbf{(B)}\ 2880\qquad\textbf{(C)}\ 3120\qquad\textbf{(D)}\ 3250\qquad\textbf{(E)}\ 3750$

Solution 1

Let vertex $A$ be any vertex, then vertex $B$ be one of the diagonal vertices to $A$, $C$ be one of the diagonal vertices to $B$, and so on. We consider cases for this problem.

In the case that $C$ has the same color as $A$, $D$ has a different color from $A$ and so $E$ has a different color from $A$ and $D$. In this case, $A$ has $6$ choices, $B$ has $5$ choices (any color but the color of $A$), $C$ has $1$ choice, $D$ has $5$ choices, and $E$ has $4$ choices, resulting in a possible of $6 \cdot 5 \cdot 1 \cdot 5 \cdot 4 = 600$ combinations.

In the case that $C$ has a different color from $A$ and $D$ has a different color from $A$, $A$ has $6$ choices, $B$ has $5$ choices, $C$ has $4$ choices (since $A$ and $B$ necessarily have different colors), $D$ has $4$ choices, and $E$ has $4$ choices, resulting in a possible of $6 \cdot 5 \cdot 4 \cdot 4 \cdot 4 = 1920$ combinations.

In the case that $C$ has a different color from $A$ and $D$ has the same color as $A$, $A$ has $6$ choices, $B$ has $5$ choices, $C$ has $4$ choices, $D$ has $1$ choice, and $E$ has $5$ choices, resulting in a possible of $6 \cdot 5 \cdot 4 \cdot 1 \cdot 5 = 600$ combinations.

Adding all those combinations up, we get $600+1920+600=\boxed{3120 \ \mathbf{(C)}}$.

Solution 2

First, notice that there can be $3$ cases. One with all vertices painted different colors, one with one pair of adjacent vertices painted the same color and the final one with two pairs of adjacent vertices painted two different colors.

Case $1$: There are $6!$ ways of assigning each vertex a different color. $6! = 720$

Case $2$: There are $\frac {6!}{2!} * 5$ ways. After picking four colors, we can rotate our pentagon in $5$ ways to get different outcomes. $\frac {6!}{2} * 5 = 1800$

Case $3$: There are $\frac {\frac {6!}{3!} * 10}{2!}$ ways of arranging the final case. We can pick $3$ colors for our pentagon. There are $5$ spots for the first pair of colors. Then, there are $2$ possible ways we can put the final pair in the last $3$ spaces. But because the two pairs are indistinguishable, we divide by $2!$. $\frac {\frac {6!}{3!} * 10}{2!} = 600$

Adding all the possibilities up, we get $720+1800+600=\boxed{3120 \ \mathbf{(C)}}$

~ZericHang

See Also

2011 AMC 10A (Problems • Answer Key • Resources)
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Problem 21 		Followed by
Problem 23
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All AMC 10 Problems and Solutions

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