Difference between revisions of "2018 AMC 8 Problems/Problem 16"
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| − | ==Problem== | + | == Problem == |
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? | 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? | ||
<math>\textbf{(A) }1440\qquad\textbf{(B) }2880\qquad\textbf{(C) }5760\qquad\textbf{(D) }182,440\qquad \textbf{(E) }362,880</math> | <math>\textbf{(A) }1440\qquad\textbf{(B) }2880\qquad\textbf{(C) }5760\qquad\textbf{(D) }182,440\qquad \textbf{(E) }362,880</math> | ||
| − | =Solution= | + | == Solution 1 == |
To solve this, treat the two Arabic books as one unit and the four Spanish books as another unit. Along with the three German books, you now have five units to arrange. | To solve this, treat the two Arabic books as one unit and the four Spanish books as another unit. Along with the three German books, you now have five units to arrange. | ||
You can arrange these five units in <math>5!</math> ways. Within the Arabic block, the two books can be arranged in <math>2!</math> ways, and within the Spanish block, the four books can be arranged in <math>4!</math> ways. | You can arrange these five units in <math>5!</math> ways. Within the Arabic block, the two books can be arranged in <math>2!</math> ways, and within the Spanish block, the four books can be arranged in <math>4!</math> ways. | ||
| − | Multiplying all these possibilities gives the total number of arrangements which is <math>\boxed{\textbf{(C) } | + | Multiplying all these possibilities gives the total number of arrangements which is <math>\boxed{\textbf{(C) }5760}.</math> ways to arrange the nine books while keeping the Arabic and Spanish books together. |
| − | ==Video Solution | + | == Video Solution 1 == |
https://youtu.be/HqeNT9ctfUo | https://youtu.be/HqeNT9ctfUo | ||
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~EzLx | ~EzLx | ||
| − | ==See Also== | + | == See Also == |
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{{AMC8 box|year=2018|num-b=15|num-a=17}} | {{AMC8 box|year=2018|num-b=15|num-a=17}} | ||
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{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 16:02, 29 January 2025
Problem
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?
Solution 1
To solve this, treat the two Arabic books as one unit and the four Spanish books as another unit. Along with the three German books, you now have five units to arrange.
You can arrange these five units in 
ways. Within the Arabic block, the two books can be arranged in 
ways, and within the Spanish block, the four books can be arranged in 
ways.
Multiplying all these possibilities gives the total number of arrangements which is 
ways to arrange the nine books while keeping the Arabic and Spanish books together.
Video Solution 1
~EzLx
See Also
| 2018 AMC 8 (Problems • Answer Key • Resources) | ||
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
