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

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==Problem==
 
==Problem==
  
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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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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>\textbf{(A) }   42   \qquad       \textbf{(B) }   49   \qquad   \textbf{(C) }   56   \qquad   \textbf{(D) } 63 \qquad \textbf{(E) }   84 </math>
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<math>
 +
\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===
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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.
 
  
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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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There are 3 ways to do so:
 
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3, 2+1, and 1+1+1
 
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There are <math>7</math> for Case 1, <math>7\cdot 6 = 42</math> for Case 2, and <math>\frac{7\cdot 6\cdot 5}{3!} = 35</math> for Case 3.
 
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Therefore, the answer is <math>7+42+35 = \boxed {\textbf{(E) } 84}</math>
 
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===Solution 2===
 
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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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===Solution 3===
 
 
Add possibilities. There are <math>3</math> ways to sum to <math>10</math>, listed below.
 
Add possibilities. There are <math>3</math> ways to sum to <math>10</math>, listed below.
  
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Add up the possibilities: <math>35+42+7=\boxed{\textbf{(E) } 84}</math>.
 
Add up the possibilities: <math>35+42+7=\boxed{\textbf{(E) } 84}</math>.
  
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Thus we have repeated Solution 1 exactly, but with less explanation.
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===Solution 2===
 +
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)}}</math>
  
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===Solution 4 (overkill)===
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===Solution 3 (overkill)===
 
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>
 
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>
  
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===Solution 5 (Stars and Bars)===
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===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 <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>
 
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>
  
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===Solution 6 (Solution 5 but more clearer and compact) ===
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=== Solution 5 (Similar to above, using number separation) ===
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Assume each die has value 1. Then we have <math>10-(1 \cdot 7)=3</math> left. This is to be split among 7 die. By stars and bars, we have <math>\binom{3+7-1}{3}=\binom{9}{3}=\boxed{84}.</math> ~mathboy282
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We can use number separation for this problem. If we set each of the dice value to <math>D\{a, b, c, d, e, f, g, h\}</math>, we can say
 +
<math>D = 10</math> and each of <math>D</math>'s elements are larger than <math>0</math>. Using the positive number separation formula, which is <math>\dbinom{n-1}{r-1}</math>, we can make the following equations.
 +
<cmath>\begin{align*}
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D &= 10 \\
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a+b+c+d+e+f+g &= 10 \\
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\dbinom{10-1}{7-1} &= \\
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\dbinom{9}{6} &= \\
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\dbinom{9}{3} &= \\
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\dfrac{9 \cdot 8 \cdot 7}{3 \cdot 2 \cdot 1} &= \\
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12 \cdot 7 &= \boxed{\textbf{(B)}84} \\
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\end{align*}</cmath>
  
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==Video Solution==
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Note: We are unable to use non-negative number separations due to the fact that the dice *must* be larger than <math>0</math> or positive.
 +
 
 +
~ Wiselion =)
 +
 
 +
==Video Solution (HOW TO THINK CREATIVELY!)==
 +
https://youtu.be/gTpg8yInCCY
 +
 
 +
~Education, the Study of Everything
 +
 
 +
 
 +
 
 +
==Video Solution 1==
 
https://youtu.be/HVn1WV80ZIU
 
https://youtu.be/HVn1WV80ZIU
  
 
~savannahsolver
 
~savannahsolver
 +
 +
== Video Solution by OmegaLearn==
 +
https://youtu.be/5UojVH4Cqqs?t=5381
 +
 +
~ pi_is_3.14
  
 
== See Also ==
 
== See Also ==

Revision as of 19:57, 7 October 2023

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 $\dbinom{9}{3}=\boxed{\textbf{(E)}}$

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 $\binom{7+3-1}{7-1}=\binom{9}{6}=\boxed{84}$

Solution 5 (Similar to above, using number separation)

We can use number separation for this problem. If we set each of the dice value to $D\{a, b, c, d, e, f, g, h\}$, we can say $D = 10$ and each of $D$'s elements are larger than $0$. Using the positive number separation formula, which is $\dbinom{n-1}{r-1}$, we can make the following equations. \begin{align*} D &= 10 \\ a+b+c+d+e+f+g &= 10 \\ \dbinom{10-1}{7-1} &= \\ \dbinom{9}{6} &= \\ \dbinom{9}{3} &= \\ \dfrac{9 \cdot 8 \cdot 7}{3 \cdot 2 \cdot 1} &= \\ 12 \cdot 7 &= \boxed{\textbf{(B)}84} \\ \end{align*}

Note: We are unable to use non-negative number separations due to the fact that the dice *must* be larger than $0$ or positive.

~ Wiselion =)

Video Solution (HOW TO THINK CREATIVELY!)

https://youtu.be/gTpg8yInCCY

~Education, the Study of Everything


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 (Problems • Answer Key • Resources)
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Problem 10 		Followed by
Problem 12
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All AMC 10 Problems and Solutions

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