Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2025 SSMO Speed Round Problems/Problem 3

Problem

Anna is buying different types of cheese from the local supermarket. Let $x,$ $y,$ and $z$ be the number of pieces of blue, cheddar, and mozzarella cheese, respectively, that Anna buys. She can buy any nonnegative integer number of each type, but the total number of pieces must be at most 12. How many different combinations $(x, y, z)$ of cheese can Anna buy? (Anna is allowed to buy 0 pieces of cheese.)

Solution

We seek the number of ordered pairs $(x,y,z)$ of nonnegative integers such that $0\le x+y+z\le 12$. Let $0\le n \le 12$ be some integer. By stars and bars, there are $\tbinom{n+2}{2}$ ordered pairs $(x,y,z)$ of nonnegative integers satisfying $x+y+z = n$. Thus, the answer is the sum \[\tbinom{2}{2} + \tbinom{3}{2} + \cdots + \tbinom{14}{2}.\] By the hockey stick identity, this is equal to $\tbinom{15}{3} = \boxed{455}$.

~Sedro

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2025_SSMO_Speed_Round_Problems/Problem_3&oldid=260145"