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Difference between revisions of "2014 AMC 10B Problems/Problem 24"

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{{duplicate|[[2014 AMC 12B Problems|2014 AMC 12B #18]] and [[2014 AMC 10B Problems|2014 AMC 10B #24]]}}
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==Problem==
 
==Problem==
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The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is ''bad'' if it is not true that for every <math>n</math> from <math>1</math> to <math>15</math> one can find a subset of the numbers that appear consecutively on the circle that sum to <math>n</math>. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?
+
The numbers <math>1, 2, 3, 4, 5</math> are to be arranged in a circle. An arrangement is <math>\textit{bad}</math> if it is not true that for every <math>n</math> from <math>1</math> to <math>15</math> one can find a subset of the numbers that appear consecutively on the circle that sum to <math>n</math>. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?
  
 
<math> \textbf {(A) } 1 \qquad \textbf {(B) } 2 \qquad \textbf {(C) } 3 \qquad \textbf {(D) } 4 \qquad \textbf {(E) } 5 </math>
 
<math> \textbf {(A) } 1 \qquad \textbf {(B) } 2 \qquad \textbf {(C) } 3 \qquad \textbf {(D) } 4 \qquad \textbf {(E) } 5 </math>
  
 
==Solution==
 
==Solution==
 +
 +
We see that there are <math>5!</math> total ways to arrange the numbers. However, we can always rotate these numbers so that, for example, the number <math>1</math> is always at the top of the circle. Thus, there are only <math>4!</math> ways under rotation. Every case has exactly <math>1</math> reflection, so that gives us only <math>4!/2</math>, or <math>12</math> cases, which is not difficult to list out. We systematically list out all <math>12</math> cases.
 +
 +
Now, we must examine if they satisfy the conditions. We can see that by choosing one number at a time, we can always obtain subsets with sums <math>1, 2, 3, 4,</math> and <math>5</math>. By choosing the full circle, we can obtain <math>15</math>. By choosing everything except for <math>1, 2, 3, 4,</math> and <math>5</math>, we can obtain subsets with sums of <math>10, 11, 12, 13,</math> and <math>14</math>.
 +
 +
This means that we now only need to check for <math>6, 7, 8,</math> and <math>9</math>. However, once we have found a set summing to <math>6</math>, we can choose everything else and obtain a set summing to <math>9</math>, and similarly for <math>7</math> and <math>8</math>. Thus, we only need to check each case for whether or not we can obtain <math>6</math> or <math>7</math>.
 +
 +
We can make <math>6</math> by having <math>4, 2</math>, or <math>3, 2, 1</math>, or <math>5, 1</math>. We can start with the group of three. To separate <math>3, 2, 1</math> from each other, they must be grouped two together and one separate, like this.
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<asy>
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draw(circle((0, 0), 5));
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pair O, A, B, C, D, E;
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O=origin;
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A=(0, 5);
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B=rotate(72)*A;
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C=rotate(144)*A;
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D=rotate(216)*A;
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E=rotate(288)*A;
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label("$x$", A, N);
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label("$y$", C, SW);
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label("$z$", D, SE);
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</asy>
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 +
Now, we note that <math>x</math> is next to both blank spots, so we can't have a number from one of the pairs. So since we can't have <math>1</math>, because it is part of the <math>5, 1</math> pair, and we can't have <math>2</math> there, because it's part of the <math>4, 2</math> pair, we must have <math>3</math> inserted into the <math>x</math> spot. We can insert <math>1</math> and <math>2</math> in <math>y</math> and <math>z</math> interchangeably, since reflections are considered the same.
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<asy>
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draw(circle((0, 0), 5));
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pair O, A, B, C, D, E;
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O=origin;
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A=(0, 5);
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B=rotate(72)*A;
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C=rotate(144)*A;
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D=rotate(216)*A;
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E=rotate(288)*A;
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label("$3$", A, N);
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label("$2$", C, SW);
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label("$1$", D, SE);
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</asy>
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We have <math>4</math> and <math>5</math> left to insert. We can't place the <math>4</math> next to the <math>2</math> or the <math>5</math> next to the <math>1</math>, so we must place <math>4</math> next to the <math>1</math> and <math>5</math> next to the <math>2</math>.
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<asy>
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draw(circle((0, 0), 5));
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pair O, A, B, C, D, E;
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O=origin;
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A=(0, 5);
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B=rotate(72)*A;
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C=rotate(144)*A;
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D=rotate(216)*A;
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E=rotate(288)*A;
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label("$3$", A, N);
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label("$5$", B, NW);
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label("$2$", C, SW);
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label("$1$", D, SE);
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label("$4$", E, NE);
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</asy>
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This is the only solution to make <math>6</math> "bad."
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 +
Next we move on to <math>7</math>, which can be made by <math>3, 4</math>, or <math>5, 2</math>, or <math>4, 2, 1</math>. We do this the same way as before. We start with the three group. Since we can't have <math>4</math> or <math>2</math> in the top slot, we must have one there, and <math>4</math> and <math>2</math> are next to each other on the bottom. When we have <math>3</math> and <math>5</math> left to insert, we place them such that we don't have the two pairs adjacent.
 +
 +
<asy>
 +
draw(circle((0, 0), 5));
 +
pair O, A, B, C, D, E;
 +
O=origin;
 +
A=(0, 5);
 +
B=rotate(72)*A;
 +
C=rotate(144)*A;
 +
D=rotate(216)*A;
 +
E=rotate(288)*A;
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label("$1$", A, N);
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label("$3$", B, NW);
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label("$2$", C, SW);
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label("$4$", D, SE);
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label("$5$", E, NE);
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</asy>
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This is the only solution to make <math>7</math> "bad."
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We've covered all needed cases, and the two examples we found are distinct, therefore the answer is <math>\boxed{\textbf {(B) }2}</math>.
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 +
Note by Williamgolly:
 +
Notice how we took care of the case of 3 numbers first. That is because that case is the most severe and we usually take care of the most severe cases first in counting problems.
 +
 +
==Video Solution==
 +
https://youtu.be/45TQEV8OjRk
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2014|ab=B|num-b=23|num-a=25}}
 
{{AMC10 box|year=2014|ab=B|num-b=23|num-a=25}}
 +
 +
{{AMC12 box|year=2014|ab=B|num-b=17|num-a=19}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 20:43, 5 October 2020

The following problem is from both the 2014 AMC 12B #18 and 2014 AMC 10B #24, so both problems redirect to this page.

Problem

The numbers $1, 2, 3, 4, 5$ are to be arranged in a circle. An arrangement is $\textit{bad}$ if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?

$\textbf {(A) } 1 \qquad \textbf {(B) } 2 \qquad \textbf {(C) } 3 \qquad \textbf {(D) } 4 \qquad \textbf {(E) } 5$

Solution

We see that there are $5!$ total ways to arrange the numbers. However, we can always rotate these numbers so that, for example, the number $1$ is always at the top of the circle. Thus, there are only $4!$ ways under rotation. Every case has exactly $1$ reflection, so that gives us only $4!/2$, or $12$ cases, which is not difficult to list out. We systematically list out all $12$ cases.

Now, we must examine if they satisfy the conditions. We can see that by choosing one number at a time, we can always obtain subsets with sums $1, 2, 3, 4,$ and $5$. By choosing the full circle, we can obtain $15$. By choosing everything except for $1, 2, 3, 4,$ and $5$, we can obtain subsets with sums of $10, 11, 12, 13,$ and $14$.

This means that we now only need to check for $6, 7, 8,$ and $9$. However, once we have found a set summing to $6$, we can choose everything else and obtain a set summing to $9$, and similarly for $7$ and $8$. Thus, we only need to check each case for whether or not we can obtain $6$ or $7$.

We can make $6$ by having $4, 2$, or $3, 2, 1$, or $5, 1$. We can start with the group of three. To separate $3, 2, 1$ from each other, they must be grouped two together and one separate, like this.

[asy] draw(circle((0, 0), 5)); pair O, A, B, C, D, E; O=origin; A=(0, 5); B=rotate(72)*A; C=rotate(144)*A; D=rotate(216)*A; E=rotate(288)*A; label("$x$", A, N); label("$y$", C, SW); label("$z$", D, SE); [/asy]

Now, we note that $x$ is next to both blank spots, so we can't have a number from one of the pairs. So since we can't have $1$, because it is part of the $5, 1$ pair, and we can't have $2$ there, because it's part of the $4, 2$ pair, we must have $3$ inserted into the $x$ spot. We can insert $1$ and $2$ in $y$ and $z$ interchangeably, since reflections are considered the same.

[asy] draw(circle((0, 0), 5)); pair O, A, B, C, D, E; O=origin; A=(0, 5); B=rotate(72)*A; C=rotate(144)*A; D=rotate(216)*A; E=rotate(288)*A; label("$3$", A, N); label("$2$", C, SW); label("$1$", D, SE); [/asy]

We have $4$ and $5$ left to insert. We can't place the $4$ next to the $2$ or the $5$ next to the $1$, so we must place $4$ next to the $1$ and $5$ next to the $2$.

[asy] draw(circle((0, 0), 5)); pair O, A, B, C, D, E; O=origin; A=(0, 5); B=rotate(72)*A; C=rotate(144)*A; D=rotate(216)*A; E=rotate(288)*A; label("$3$", A, N); label("$5$", B, NW); label("$2$", C, SW); label("$1$", D, SE); label("$4$", E, NE); [/asy]

This is the only solution to make $6$ "bad."

Next we move on to $7$, which can be made by $3, 4$, or $5, 2$, or $4, 2, 1$. We do this the same way as before. We start with the three group. Since we can't have $4$ or $2$ in the top slot, we must have one there, and $4$ and $2$ are next to each other on the bottom. When we have $3$ and $5$ left to insert, we place them such that we don't have the two pairs adjacent.

[asy] draw(circle((0, 0), 5)); pair O, A, B, C, D, E; O=origin; A=(0, 5); B=rotate(72)*A; C=rotate(144)*A; D=rotate(216)*A; E=rotate(288)*A; label("$1$", A, N); label("$3$", B, NW); label("$2$", C, SW); label("$4$", D, SE); label("$5$", E, NE); [/asy]

This is the only solution to make $7$ "bad."

We've covered all needed cases, and the two examples we found are distinct, therefore the answer is $\boxed{\textbf {(B) }2}$.

Note by Williamgolly: Notice how we took care of the case of 3 numbers first. That is because that case is the most severe and we usually take care of the most severe cases first in counting problems.

Video Solution

https://youtu.be/45TQEV8OjRk

See Also

2014 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
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
2014 AMC 12B (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 12 Problems and Solutions

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