Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2009 AIME I Problems/Problem 10"

(→‎Solution)
(→‎Solution)
Line 4: Line 4:
  
 
== Solution ==
 
== Solution ==
−
Since the 5 members of each planet committee are distinct we get that the number of arrangement of sittings is in the form <math>N*(5!)^3</math> because for each M,V,E sequence we have <math>5!</math> arrangements within the Ms, Vs, and Es.
+
Since the 5 members of each planet committee are distinct we get that the number of arrangement of sittings is in the form <math>N*(5!)^3</math> because for each <math>M, V, E</math> sequence we have <math>5!</math> arrangements within the Ms, Vs, and Es.
  
−
Pretend the table only seats 3 "people", with 1 "person" from each planet. Counting clockwise, only the arrangement M, V, E satisfies the given constraints. Therefore, in the actual problem, the members must sit in cycles of M, V, E, but not necessarily with one M, one V, and one E in each cycle(for example, MMVVVE, MVVVEEE, MMMVVVEE all count as cycles). These cycles of MVE must start at seat 1, since a M is at seat 1. We simply count the number of arrangements through casework.
+
Pretend the table only seats <math>3</math> "people", with <math>1</math> "person" from each planet. Counting clockwise, only the arrangement M, V, E satisfies the given constraints. Therefore, in the actual problem, the members must sit in cycles of M, V, E, but not necessarily with one M, one V, and one E in each cycle(for example, MMVVVE, MVVVEEE, MMMVVVEE all count as cycles). These cycles of MVE must start at seat <math>1</math>, since an M is at seat <math>1</math>. We simply count the number of arrangements through casework.
  
 
1. The entire arrangement is one cycle- There is only one way to arrange this, MMMMMVVVVVEEEEE
 
1. The entire arrangement is one cycle- There is only one way to arrange this, MMMMMVVVVVEEEEE
  
−
2. Two cycles - There are 3 Ms, Vs and Es left to distribute among the existing MVEMVE. Using sticks and stones(stars and bars), we get C(4,1)=4 ways for the members of each planet. Therefore, there are <math>4^3=64</math> ways in total.
+
2. Two cycles - There are 3 Ms, Vs and Es left to distribute among the existing MVEMVE. Using sticks and stones(stars and bars), we get <math>\binom{4}{1}=4</math> ways for the members of each planet. Therefore, there are <math>4^3=64</math> ways in total.
  
−
3. Three cycles - 2 Ms, Vs, Es left, so C(4,2)=6 and 216 ways total.
+
3. Three cycles - 2 Ms, Vs, Es left, so <math>\binom{4}{2}=6</math>, making there <math>6^3=216</math> ways total.
  
−
4. Four cycles - 1 M, V, E left, each M can go to any of the four MVE cycles and likewise for V and E, 64 ways total
+
4. Four cycles - 1 M, V, E left, each M can go to any of the four MVE cycles and likewise for V and E, <math>4^3=64</math> ways total
−
 
+
−
5. Five cycles - MVEMVEMVEMVEMVE=1 way
+
5. Five cycles - MVEMVEMVEMVEMVE is the only possibility, so there is just <math>1</math> way.
  
 
Combining all these cases, we get <math>1+1+64+64+216= \boxed{346}</math>
 
Combining all these cases, we get <math>1+1+64+64+216= \boxed{346}</math>

Revision as of 17:23, 17 March 2018

Problem

The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $1$ to $15$ in clockwise order. Committee rules state that a Martian must occupy chair $1$ and an Earthling must occupy chair $15$, Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is $N(5!)^3$. Find $N$.

Solution

Since the 5 members of each planet committee are distinct we get that the number of arrangement of sittings is in the form $N*(5!)^3$ because for each $M, V, E$ sequence we have $5!$ arrangements within the Ms, Vs, and Es.

Pretend the table only seats $3$ "people", with $1$ "person" from each planet. Counting clockwise, only the arrangement M, V, E satisfies the given constraints. Therefore, in the actual problem, the members must sit in cycles of M, V, E, but not necessarily with one M, one V, and one E in each cycle(for example, MMVVVE, MVVVEEE, MMMVVVEE all count as cycles). These cycles of MVE must start at seat $1$, since an M is at seat $1$. We simply count the number of arrangements through casework.

1. The entire arrangement is one cycle- There is only one way to arrange this, MMMMMVVVVVEEEEE

2. Two cycles - There are 3 Ms, Vs and Es left to distribute among the existing MVEMVE. Using sticks and stones(stars and bars), we get $\binom{4}{1}=4$ ways for the members of each planet. Therefore, there are $4^3=64$ ways in total.

3. Three cycles - 2 Ms, Vs, Es left, so $\binom{4}{2}=6$, making there $6^3=216$ ways total.

4. Four cycles - 1 M, V, E left, each M can go to any of the four MVE cycles and likewise for V and E, $4^3=64$ ways total

5. Five cycles - MVEMVEMVEMVEMVE is the only possibility, so there is just $1$ way.

Combining all these cases, we get $1+1+64+64+216= \boxed{346}$

See also

2009 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2009_AIME_I_Problems/Problem_10&oldid=93304"