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

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: $\frac{9!}{6!\,2!\,1!} = 252, \qquad \frac{9!}{5!\,3!\,1!} = 504, \qquad \frac{9!}{4!\,3!\,2!} = 1260.$ Thus $N = 252 + 504 + 1260 = 2016,$ and the remainder modulo $1000$ is $16.$

2025 AIME I Problem 3

Solution:

Problem 3 in Other Years