Difference between revisions of "2018 AMC 8 Problems/Problem 16"
(Created page with "==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 ni...") |
m (→=Solution) |
||
| (37 intermediate revisions by 24 users not shown) | |||
| Line 1: | Line 1: | ||
| − | ==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= | ||
| + | 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 5! ways. Within the Arabic block, the two books can be arranged in 2! ways, and within the Spanish block, the four books can be arranged in 4! ways. | ||
| + | |||
| + | Multiplying all these possibilities gives the total number of arrangements as: | ||
| + | |||
| + | . | ||
| + | |||
| + | This results in 5,760 ways to arrange the nine books while keeping the Arabic and Spanish books together. | ||
| + | |||
| + | ==Video Solution (CREATIVE ANALYSIS!!!)== | ||
| + | https://youtu.be/GznYT3zMh3o | ||
| + | |||
| + | ~Education, the Study of Everything | ||
| + | |||
| + | ==Video Solutions== | ||
| + | https://youtu.be/ci5Nwd1UnNE | ||
| + | |||
| + | https://youtu.be/FqyYAiyjCds | ||
| + | |||
| + | ~savannahsolver | ||
| + | |||
| + | https://www.youtube.com/watch?v=t1IARvN-JMM ~David | ||
| + | |||
| + | ==See Also== | ||
| + | {{AMC8 box|year=2018|num-b=15|num-a=17}} | ||
| + | |||
| + | {{MAA Notice}} | ||
Latest revision as of 06:26, 17 October 2024
Contents
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
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 5! ways. Within the Arabic block, the two books can be arranged in 2! ways, and within the Spanish block, the four books can be arranged in 4! ways.
Multiplying all these possibilities gives the total number of arrangements as:
.
This results in 5,760 ways to arrange the nine books while keeping the Arabic and Spanish books together.
Video Solution (CREATIVE ANALYSIS!!!)
~Education, the Study of Everything
Video Solutions
~savannahsolver
https://www.youtube.com/watch?v=t1IARvN-JMM ~David
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. 
