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Difference between revisions of "2017 AMC 8 Problems/Problem 10"

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<math>\textbf{(A) }\frac{1}{10}\qquad\textbf{(B) }\frac{1}{5}\qquad\textbf{(C) }\frac{3}{10}\qquad\textbf{(D) }\frac{2}{5}\qquad\textbf{(E) }\frac{1}{2}</math>
 
<math>\textbf{(A) }\frac{1}{10}\qquad\textbf{(B) }\frac{1}{5}\qquad\textbf{(C) }\frac{3}{10}\qquad\textbf{(D) }\frac{2}{5}\qquad\textbf{(E) }\frac{1}{2}</math>
  
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==Video Solution==
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==Solution 1 (combinatianol)==
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https://youtu.be/OOdK-nOzaII?t=1237
 
  
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==Solution==
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There are <math>\binom{5}{3}</math> possible groups of cards that can be selected. If <math>4</math> is the largest card selected, then the other two cards must be either <math>1</math>, <math>2</math>, or <math>3</math>, for a total <math>\binom{3}{2}</math> groups of cards. Then, the probability is just <math>{\frac{{\dbinom{3}{2}}}{{\dbinom{5}{3}}}} = \boxed{{\textbf{(C) }} {\frac{3}{10}}}</math>.
  
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There are <math>\binom{5}{3}</math> possible groups of cards that can be selected. If <math>4</math> is the largest card selected, then the other two cards must be either <math>1</math>, <math>2</math>, or <math>3</math>, for a total <math>\binom{3}{2}</math> groups of cards. Then the probability is just <math>{\frac{{\dbinom{3}{2}}}{{\dbinom{5}{3}}}} = \boxed{{\textbf{(C) }} {\frac{3}{10}}}</math>
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==Solution 2 (regular probability)==
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P (no 5)= <math>\frac{4}{5}</math> * <math>\frac{3}{4}</math> * <math>\frac{2}{3}</math> = <math>\frac{2}{5}</math>. This is the fraction of total cases with no fives.
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p (no 4 and no 5)= <math>\frac{3}{5}</math> * <math>\frac{2}{4}</math> * <math>\frac{1}{3}</math> = <math>\frac{6}{60}</math> = <math>\frac{1}{10}</math>. This is the intersection of no fours and no fives. Subtract the fraction of no fours and no fives from that of no fives. <math>\frac{2}{5} - \frac{1}{10} = \frac{3}{10} = \boxed{{\textbf{(C) }} {\frac{3}{10}}}</math>.
  
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==Solution 2 (regular probability)==
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==Video Solution (CREATIVE THINKING!!!)==
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P (no 5)= 4/5*3/4*2/3=2/5 this is the fraction of total cases with no fives.
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https://youtu.be/P-K9AEAuhNY
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p (no 4 and no 5)= 3/5*2/4*1/3= 6/60=1/10 this is the intersection of no fours and no fives. Subtract fraction of no fours and no fives from no fives.
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~Education, the Study of Everything
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==Video Solutions==
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*https://youtu.be/OOdK-nOzaII?t=1237
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*https://youtu.be/M9kj4ztWbwo
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https://youtu.be/FN9qkU62a9U
  
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<math>2/5 - 1/10 = 3/10</math> (C)
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~savannahsolver
  
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Video here:
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*https://www.youtube.com/watch?v=F935tcVcXvY  ~David
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https://youtu.be/M9kj4ztWbwo
 
  
 
==See Also:==
 
==See Also:==

Latest revision as of 23:22, 3 November 2024

Problem 10

A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected?

$\textbf{(A) }\frac{1}{10}\qquad\textbf{(B) }\frac{1}{5}\qquad\textbf{(C) }\frac{3}{10}\qquad\textbf{(D) }\frac{2}{5}\qquad\textbf{(E) }\frac{1}{2}$

Solution 1 (combinatianol)

There are $\binom{5}{3}$ possible groups of cards that can be selected. If $4$ is the largest card selected, then the other two cards must be either $1$, $2$, or $3$, for a total $\binom{3}{2}$ groups of cards. Then, the probability is just ${\frac{{\dbinom{3}{2}}}{{\dbinom{5}{3}}}} = \boxed{{\textbf{(C) }} {\frac{3}{10}}}$.

Solution 2 (regular probability)

P (no 5)= $\frac{4}{5}$ * $\frac{3}{4}$ * $\frac{2}{3}$ = $\frac{2}{5}$. This is the fraction of total cases with no fives. p (no 4 and no 5)= $\frac{3}{5}$ * $\frac{2}{4}$ * $\frac{1}{3}$ = $\frac{6}{60}$ = $\frac{1}{10}$. This is the intersection of no fours and no fives. Subtract the fraction of no fours and no fives from that of no fives. $\frac{2}{5} - \frac{1}{10} = \frac{3}{10} = \boxed{{\textbf{(C) }} {\frac{3}{10}}}$.

Video Solution (CREATIVE THINKING!!!)

https://youtu.be/P-K9AEAuhNY

~Education, the Study of Everything

Video Solutions

https://youtu.be/FN9qkU62a9U

~savannahsolver

See Also:

2017 AMC 8 (Problems • Answer Key • Resources)
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Problem 9 		Followed by
Problem 11
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