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Difference between revisions of "2014 AMC 10A Problems/Problem 17"

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<math> \textbf{(A)}\ \dfrac16\qquad\textbf{(B)}\ \dfrac{13}{72}\qquad\textbf{(C)}\ \dfrac7{36}\qquad\textbf{(D)}\ \dfrac5{24}\qquad\textbf{(E)}\ \dfrac29 </math>
 
<math> \textbf{(A)}\ \dfrac16\qquad\textbf{(B)}\ \dfrac{13}{72}\qquad\textbf{(C)}\ \dfrac7{36}\qquad\textbf{(D)}\ \dfrac5{24}\qquad\textbf{(E)}\ \dfrac29 </math>
  
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==Solution 1 (Clean Counting)==
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==Solution 1==
  
 
First, we note that there are <math>1, 2, 3, 4,</math> and <math>5</math> ways to get sums of <math>2, 3, 4, 5, 6</math> respectively--this is not too hard to see. With any specific sum, there is exactly one way to attain it on the other die. This means that the probability that two specific dice have the same sum as the other is <cmath>\dfrac16 \left( \dfrac{1+2+3+4+5}{36}\right) = \dfrac{5}{72}.</cmath> Since there are <math>\dbinom31</math> ways to choose which die will be the one with the sum of the other two, our answer is <math>3 \cdot \dfrac{5}{72} = \boxed{\textbf{(D)} \: \dfrac{5}{24}}</math>.
 
First, we note that there are <math>1, 2, 3, 4,</math> and <math>5</math> ways to get sums of <math>2, 3, 4, 5, 6</math> respectively--this is not too hard to see. With any specific sum, there is exactly one way to attain it on the other die. This means that the probability that two specific dice have the same sum as the other is <cmath>\dfrac16 \left( \dfrac{1+2+3+4+5}{36}\right) = \dfrac{5}{72}.</cmath> Since there are <math>\dbinom31</math> ways to choose which die will be the one with the sum of the other two, our answer is <math>3 \cdot \dfrac{5}{72} = \boxed{\textbf{(D)} \: \dfrac{5}{24}}</math>.
  
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--happiface
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==Solution 2 (Bashy)==
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<math>(1, 2, 3); (1, 3, 4); (1, 4, 5); (1, 5, 6); (2, 3, 5); (2, 4, 6)</math> have <math>6</math> ways to rearrange them for a total of <math>36</math> ways. <math>(1, 1, 2); (2, 2, 4); (3, 3, 6)</math> have <math>3</math> ways to rearrange them for a total of <math>9</math> ways. Adding them up, we get <math>45</math> ways. We have to divide this values by <math>6^3</math> because there are 3 dice. <math>\dfrac{45}{216}=\boxed{\dfrac{5}{24}}</math>.
  
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==Solution 2 (Casework)==
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~MathFun1000
  
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Since there are <math>6</math> possible values for the number on each dice, there are <math>6^3=216</math> total possible rolls.
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==Solution 3 (Summary of Solution 2)==
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Start by listing all possible sums. Then find the ways to arrange them and sum them up and divide by <math>6^3</math> for 3 dice. <math>\dfrac{45}{216}</math> is <math>\boxed{\textbf{(D)} \: \dfrac{5}{24}}</math>.
  
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The possible results of the 3 dice such that the sum of the values of two of the die is equal to the value of the third die are, without considering the order of the die,  <cmath>(1, 1, 2), (1, 2, 3), (1, 3, 4), (1, 4, 5), (1, 5, 6), (2, 2, 4), (2, 3, 5), (2, 4, 6), (3, 3, 6)</cmath>
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-aopspandy
  
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There are <math>\frac{3!}{2}=3</math> ways to order the first, sixth, and ninth results, and there are <math>3!=6</math> ways to order the other results.
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==Solution 4 (enumeration and casework)==
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There are <math>3</math> ways to pick the dice which is the sum. From here proceed with "casework":
 +
*If the sum is 1, there are <math>0</math> ways because the minimum sum is <math>2</math>.
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*If the sum is 2, there is <math>1</math> way (namely <math>(1, 1)</math>).
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*If the sum is 3, there are <math>2</math> ways (namely <math>(1, 2)</math> and <math>(2, 1)</math>).
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*If the sum is 4, there are <math>3</math> ways (namely <math>(1, 3)</math>, <math>(3, 1)</math>, and <math>(2, 2)</math>).
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We start noticing that for a sum of <math>n</math> there are <math>n-1</math> cases, but let's keep going to verify.
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*If the sum is 5, there are <math>4</math> ways (namely <math>(1, 4)</math>, <math>(4, 1)</math>, <math>(2, 3)</math>, and <math>(3, 2)</math>).
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*If the sum is 6, there are <math>5</math> ways (namely <math>(1, 5)</math>, <math>(5, 1)</math>, <math>(2, 4)</math>, <math>(4, 2)</math>, and <math>(3, 3)</math>).
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We have exhausted all cases because there is no way a dice can roll a <math>7</math>. Summing, there are 15 ways here.
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To summarize: there are <math>3</math> ways to pick the dice that is the sum, and <math>15</math> ways to achieve the sum, for a total of <math>45</math> configurations. There are <math>6\cdot6\cdot6=216</math> total configurations, and the probability is <math>\frac{45}{216}=\frac{5}{24} \Longrightarrow \boxed{\textbf{(D)} \frac{5}{24}}</math>.
  
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Therefore, there are a total of <math>3\times3+6\times6=45</math> ways to roll the dice such that 2 of the dice sum to the other die, so our answer is <cmath>\frac{45}{216}=\boxed{\textbf{(D)} \ \frac{5}{24}}</cmath>
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~JH. L
  
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(Solution by bestwillcui1)
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==Video Solution by OmegaLearn==
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https://youtu.be/5UojVH4Cqqs?t=702
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~ pi_is_3.14
  
 
==See Also==
 
==See Also==
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{{AMC10 box|year=2014|ab=A|num-b=16|num-a=18}}
 
{{AMC10 box|year=2014|ab=A|num-b=16|num-a=18}}
 
{{MAA Notice}}
 
{{MAA Notice}}
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 +
[[Category: Introductory Combinatorics Problems]]

Latest revision as of 18:34, 11 September 2024

Problem

Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?

$\textbf{(A)}\ \dfrac16\qquad\textbf{(B)}\ \dfrac{13}{72}\qquad\textbf{(C)}\ \dfrac7{36}\qquad\textbf{(D)}\ \dfrac5{24}\qquad\textbf{(E)}\ \dfrac29$


Solution 1

First, we note that there are $1, 2, 3, 4,$ and $5$ ways to get sums of $2, 3, 4, 5, 6$ respectively--this is not too hard to see. With any specific sum, there is exactly one way to attain it on the other die. This means that the probability that two specific dice have the same sum as the other is \[\dfrac16 \left( \dfrac{1+2+3+4+5}{36}\right) = \dfrac{5}{72}.\] Since there are $\dbinom31$ ways to choose which die will be the one with the sum of the other two, our answer is $3 \cdot \dfrac{5}{72} = \boxed{\textbf{(D)} \: \dfrac{5}{24}}$.

Solution 2 (Bashy)

$(1, 2, 3); (1, 3, 4); (1, 4, 5); (1, 5, 6); (2, 3, 5); (2, 4, 6)$ have $6$ ways to rearrange them for a total of $36$ ways. $(1, 1, 2); (2, 2, 4); (3, 3, 6)$ have $3$ ways to rearrange them for a total of $9$ ways. Adding them up, we get $45$ ways. We have to divide this values by $6^3$ because there are 3 dice. $\dfrac{45}{216}=\boxed{\dfrac{5}{24}}$.

~MathFun1000

Solution 3 (Summary of Solution 2)

Start by listing all possible sums. Then find the ways to arrange them and sum them up and divide by $6^3$ for 3 dice. $\dfrac{45}{216}$ is $\boxed{\textbf{(D)} \: \dfrac{5}{24}}$.

-aopspandy

Solution 4 (enumeration and casework)

There are $3$ ways to pick the dice which is the sum. From here proceed with "casework":

  • If the sum is 1, there are $0$ ways because the minimum sum is $2$.
  • If the sum is 2, there is $1$ way (namely $(1, 1)$).
  • If the sum is 3, there are $2$ ways (namely $(1, 2)$ and $(2, 1)$).
  • If the sum is 4, there are $3$ ways (namely $(1, 3)$, $(3, 1)$, and $(2, 2)$).

We start noticing that for a sum of $n$ there are $n-1$ cases, but let's keep going to verify.

  • If the sum is 5, there are $4$ ways (namely $(1, 4)$, $(4, 1)$, $(2, 3)$, and $(3, 2)$).
  • If the sum is 6, there are $5$ ways (namely $(1, 5)$, $(5, 1)$, $(2, 4)$, $(4, 2)$, and $(3, 3)$).

We have exhausted all cases because there is no way a dice can roll a $7$. Summing, there are 15 ways here. To summarize: there are $3$ ways to pick the dice that is the sum, and $15$ ways to achieve the sum, for a total of $45$ configurations. There are $6\cdot6\cdot6=216$ total configurations, and the probability is $\frac{45}{216}=\frac{5}{24} \Longrightarrow \boxed{\textbf{(D)} \frac{5}{24}}$.

~JH. L

Video Solution by OmegaLearn

https://youtu.be/5UojVH4Cqqs?t=702

~ pi_is_3.14

See Also

2014 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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

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