Difference between revisions of "2018 AMC 10A Problems/Problem 11"

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When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be written as <cmath>\frac{n}{6^7},</cmath>where <math>n</math> is a positive integer. What is <math>n</math>?
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==Problem==
  
<math>\textbf{(A) }   42   \qquad       \textbf{(B) }   49   \qquad   \textbf{(C) }   56   \qquad   \textbf{(D) } 63 \qquad \textbf{(E) }   84 </math>
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When <math>7</math> fair standard <math>6</math>-sided dice are thrown, the probability that the sum of the numbers on the top faces is <math>10</math> can be written as <cmath>\frac{n}{6^{7}},</cmath> where <math>n</math> is a positive integer. What is <math>n</math>?
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<math>
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\textbf{(A) }42\qquad
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\textbf{(B) }49\qquad
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\textbf{(C) }56\qquad
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\textbf{(D) }63\qquad
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\textbf{(E) }84\qquad
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</math>
  
 
== Solutions ==
 
== Solutions ==
 
===Solution 1===
 
===Solution 1===
The minimum number that can be shown on the face of a die is 1, so the least possible sum of the top faces of the 7 dies is 7.
 
  
In order for the sum to be exactly 10, 1 to 3 dices' number on the top face must be increased by a total of 3.  
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Add possibilities. There are <math>3</math> ways to sum to <math>10</math>, listed below.
 
 
There are 3 ways to do so:
 
3, 2+1, and 1+1+1
 
 
 
There are <math>\dbinom {7}{1}</math> for Case 1, <math>7*6 = 42</math> for Case 2, and <math>\dbinom {7}{3}</math> for Case 3.
 
  
Therefore, the answer is <math>7+42+35 = \boxed {\textbf{(E) } 84}</math>
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<cmath>4,1,1,1,1,1,1: 7</cmath>
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<cmath>3,2,1,1,1,1,1: 42</cmath>
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<cmath>2,2,2,1,1,1,1: 35.</cmath>
  
~PancakeMonster2004
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Add up the possibilities: <math>35+42+7=\boxed{\textbf{(E) } 84}</math>.
  
 
===Solution 2===
 
===Solution 2===
Rolling a sum of 10 with 7 dice can be represented with stars and bars, with 10 stars and 6 bars. Each star represents one of the dots on the die's faces and the bars represent separation between different dice. However, we must note that each die must have at least one dot on a face, so there must already be 7 stars predetermined. We are left with 3 stars and 6 bars, which we can rearrange in <math>\dbinom{9}{3}=\boxed{\textbf{(E) } 84}</math> ways.
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Rolling a sum of <math>10</math> with 7 dice can be represented with stars and bars, with 10 stars and 6 bars. Each star represents one of the dots on the dices' faces and the bars represent separation between different dice. However, we must note that each die must have at least one dot on a face, so there must already be 7 stars predetermined. We are left with 3 stars and 6 bars, which we can rearrange in <math>\dbinom{9}{3}=\boxed{\textbf{(E) } 84}</math> ways.
  
~RegularHexagon
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===Solution 3 (overkill)===
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We can use generating functions, where <math>(x+x^2+...+x^6)</math> is the function for each die. We want to find the coefficient of <math>x^{10}</math> in <math>(x+x^2+...+x^6)^7</math>, which is the coefficient of <math>x^3</math> in <math>\left(\frac{1-x^7}{1-x}\right)^7</math>. This evaluates to <math>\dbinom{-7}{3} \cdot (-1)^3=\boxed{\textbf{(E) } 84}</math>
  
===Solution 3===
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===Solution 4 (Stars and Bars)===
Add possibilities. There are <math>3</math> ways to sum to <math>10</math>, listed below.
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If we let each number take its minimum value of 1, we will get 7 as the minimum sum. So we can do <math>10</math> - <math>7</math> = <math>3</math> to find the number of balls we need to distribute to get three more added to the minimum to get 10, so the problem is asking how many ways can you put <math>3</math> balls into <math>7</math> boxes. From there we get <math>\binom{7+3-1}{7-1}=\binom{9}{6}=\boxed{84}</math>
  
<math>4,1,1,1,1,1,1: 7</math>
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==Video Solution 1==
<math>3,2,1,1,1,1,1: 42</math>
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https://youtu.be/HVn1WV80ZIU
<math>2,2,2,1,1,1,1: 35.</math>
 
  
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~savannahsolver
  
Add up the possibilities: <math>35+42+7=\boxed{84}.</math> The answer is <math>\textbf{(E)}</math>.
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== Video Solution by OmegaLearn==
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https://youtu.be/5UojVH4Cqqs?t=5381
  
~kevinmathz
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~ pi_is_3.14
  
 
== See Also ==
 
== See Also ==
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{{AMC10 box|year=2018|ab=A|num-b=10|num-a=12}}
 
{{AMC10 box|year=2018|ab=A|num-b=10|num-a=12}}
 
{{MAA Notice}}
 
{{MAA Notice}}
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[[Category:Introductory Probability Problems]]

Latest revision as of 22:32, 20 November 2022

Problem

When $7$ fair standard $6$-sided dice are thrown, the probability that the sum of the numbers on the top faces is $10$ can be written as \[\frac{n}{6^{7}},\] where $n$ is a positive integer. What is $n$?

$\textbf{(A) }42\qquad \textbf{(B) }49\qquad \textbf{(C) }56\qquad \textbf{(D) }63\qquad \textbf{(E) }84\qquad$

Solutions

Solution 1

Add possibilities. There are $3$ ways to sum to $10$, listed below.

\[4,1,1,1,1,1,1: 7\] \[3,2,1,1,1,1,1: 42\] \[2,2,2,1,1,1,1: 35.\]

Add up the possibilities: $35+42+7=\boxed{\textbf{(E) } 84}$.

Solution 2

Rolling a sum of $10$ with 7 dice can be represented with stars and bars, with 10 stars and 6 bars. Each star represents one of the dots on the dices' faces and the bars represent separation between different dice. However, we must note that each die must have at least one dot on a face, so there must already be 7 stars predetermined. We are left with 3 stars and 6 bars, which we can rearrange in $\dbinom{9}{3}=\boxed{\textbf{(E) } 84}$ ways.

Solution 3 (overkill)

We can use generating functions, where $(x+x^2+...+x^6)$ is the function for each die. We want to find the coefficient of $x^{10}$ in $(x+x^2+...+x^6)^7$, which is the coefficient of $x^3$ in $\left(\frac{1-x^7}{1-x}\right)^7$. This evaluates to $\dbinom{-7}{3} \cdot (-1)^3=\boxed{\textbf{(E) } 84}$

Solution 4 (Stars and Bars)

If we let each number take its minimum value of 1, we will get 7 as the minimum sum. So we can do $10$ - $7$ = $3$ to find the number of balls we need to distribute to get three more added to the minimum to get 10, so the problem is asking how many ways can you put $3$ balls into $7$ boxes. From there we get $\binom{7+3-1}{7-1}=\binom{9}{6}=\boxed{84}$

Video Solution 1

https://youtu.be/HVn1WV80ZIU

~savannahsolver

Video Solution by OmegaLearn

https://youtu.be/5UojVH4Cqqs?t=5381

~ pi_is_3.14

See Also

2018 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
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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