2025 AIME I Problem 3

Question

The

9

members of a baseball team went to an ice-cream parlor after their game. Each player had a single scoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was greater than the number of players who chose strawberry. Let

N

be the number of different assignments of flavors to players that meet these conditions. Find the remainder when

N

is divided by

1000.

Answer

Answer: 16

Concepts:multiset permutations casework

Difficulty rating: 2180

Small hint:

First find every triple of flavor counts: strictly decreasing positive integers summing to

9

Big hint:

For counts

(c, v, s),

the number of ways to assign flavors to the nine distinct players is

\frac{9!}{c!\,v!\,s!}

Solution:

Let

c \gt v \gt s \ge 1

be the numbers of players choosing chocolate, vanilla, and strawberry, with

c + v + s = 9.

Checking small values of

s

shows the only possibilities are

(6, 2, 1),

(5, 3, 1),

and

(4, 3, 2).

Since the players are distinct, each triple of counts contributes a multinomial coefficient:

\begin{aligned} \frac{9!}{6!\,2!\,1!} &= 252, \\ \frac{9!}{5!\,3!\,1!} &= 504, \\ \frac{9!}{4!\,3!\,2!} &= 1260. \end{aligned}

Thus

N = 252 + 504 + 1260 = 2016,

and the remainder modulo

1000

is

16.