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Difference between revisions of "2025 AIME I Problems/Problem 3"

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==Problem==
 
==Problem==
 
The <math>9</math> 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 <math>N</math> be the number of different assignments of flavors to players that meet these conditions. Find the remainder when <math>N</math> is divided by <math>1000.</math>
 
The <math>9</math> 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 <math>N</math> be the number of different assignments of flavors to players that meet these conditions. Find the remainder when <math>N</math> is divided by <math>1000.</math>
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==Solution 1==
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We apply casework on the scoops the team gets.
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<u>Case 1:</u> The scoops are <math>6,2,1</math>. Then we have <math>\binom{9}{6}\cdot \binom{3}{2} = 252</math>.
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<u>Case 2:</u> The scoops are <math>5,3,1</math>. Then we have <math>\binom{9}{5}\cdot \binom{4}{3} = 504</math>.
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<u>Case 3:</u> The scoops are <math>4,3,2</math>. Then we have <math>\binom{9}{4}\cdot \binom{5}{3} = 1260</math>.
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Thus the answer is <math>252+504+1260=2\boxed{016}</math>.
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~ [[User:zhenghua|zhenghua]]
  
 
==See also==
 
==See also==

Revision as of 19:43, 13 February 2025

Problem

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 $1000.$

Solution 1

We apply casework on the scoops the team gets.

Case 1: The scoops are $6,2,1$. Then we have $\binom{9}{6}\cdot \binom{3}{2} = 252$.

Case 2: The scoops are $5,3,1$. Then we have $\binom{9}{5}\cdot \binom{4}{3} = 504$.

Case 3: The scoops are $4,3,2$. Then we have $\binom{9}{4}\cdot \binom{5}{3} = 1260$.

Thus the answer is $252+504+1260=2\boxed{016}$.

~ zhenghua

See also

2025 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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