2009 AIME I Problems/Problem 10
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 to
in clockwise order. Committee rules state that a Martian must occupy chair
and an Earthling must occupy chair
, 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
. Find
.
Solution
Since after each planet, only members of another planet can follow, we simply count the lengths of the blocks adding up to ten. We consider a few different cases:
1. One block of five people- There is only one way to arrange this so .
2. Five blocks of one person - There is also only one way to arrange this so we get .
3. Two blocks - There are two cases: and
. Each of these can be arranged two ways so we get
.
4. Three blocks - There are also two cases: and
.Each of these can be arranged three ways giving us
.
5. Four blocks - There is only one case: . This can be arranged four ways giving us
.
Combining all these cases, we get
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 |