Difference between revisions of "2018 AMC 8 Problems/Problem 16"

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==Solution==
 
==Solution==
 
Since the Arabic books and Spanish books have to be kept together, we can treat them both as just one book. That means we're trying to find the number of ways you can arrange one Arabic book, one Spanish book, and three German books, which is just <math>5</math> factorial. Now we multiply this product by <math>2!</math> because there are <math>2</math> factorial ways to arrange just the Arabic books, and <math>4!</math> ways to arrange just the Spanish books. Multiplying all these together, we have <math>2!\cdot 4!\cdot 5!=\boxed{\textbf{(C) }5760.}</math>
 
Since the Arabic books and Spanish books have to be kept together, we can treat them both as just one book. That means we're trying to find the number of ways you can arrange one Arabic book, one Spanish book, and three German books, which is just <math>5</math> factorial. Now we multiply this product by <math>2!</math> because there are <math>2</math> factorial ways to arrange just the Arabic books, and <math>4!</math> ways to arrange just the Spanish books. Multiplying all these together, we have <math>2!\cdot 4!\cdot 5!=\boxed{\textbf{(C) }5760.}</math>
 
  
 
==See Also==
 
==See Also==

Revision as of 21:49, 28 October 2020

Problem 16

Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?

$\textbf{(A) }1440\qquad\textbf{(B) }2880\qquad\textbf{(C) }5760\qquad\textbf{(D) }182,440\qquad \textbf{(E) }362,880$

Solution

Since the Arabic books and Spanish books have to be kept together, we can treat them both as just one book. That means we're trying to find the number of ways you can arrange one Arabic book, one Spanish book, and three German books, which is just $5$ factorial. Now we multiply this product by $2!$ because there are $2$ factorial ways to arrange just the Arabic books, and $4!$ ways to arrange just the Spanish books. Multiplying all these together, we have $2!\cdot 4!\cdot 5!=\boxed{\textbf{(C) }5760.}$

See Also

2018 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 15
Followed by
Problem 17
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 AJHSME/AMC 8 Problems and Solutions

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