2025 AIME II Problems/Problem 10
Problem
Sixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let 
be the number of subsets of 
chairs that could be selected. Find the remainder when is divided by 
.
Solution 1 (Casework)
Let's split the problem into a few cases:
Case 1: All 
people are sitting isolated (no person sits next to any of them): 
Case 2: 
people are isolated, 
people sit next to each other (such that each person sits next to either 
or 
other person): 
Case 3: 
people are isolated, people sit next to each other and other people sit next to each other with the groups of people not sitting next to each other (so each person still sits next to either or other person): 
Case 4: people are isolated, people are split into 
groups of people, and no groups sit next to each other: 
Case 5: groups of , no groups are sitting next to each other: 
We have 
~Mitsuihisashi14
Solution 2 (Stars-and-bars)
Let 
be the number of groups of adjacent people.
Clearly, there will be 
groups with people, and there are 
ways to select these groups.
Now, we use the stars-and-bars technique to find the number of ways to distribute the remaining chairs around the people. Since the groups are separated, we must choose 
chairs to place in between beforehand, leaving 
chairs to distribute among 
spots. Using stars-and-bars, we find that the number of ways to do this is 
.
Thus, the total number of arrangements is 
. Summing this from 
to 
yields 
.
See Also
| 2025 AIME II (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 | ||
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